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Properties

Label 2-74-37.16-c1-0-1
Degree 22
Conductor 7474
Sign 0.928+0.370i0.928 + 0.370i
Analytic cond. 0.5908920.590892
Root an. cond. 0.7686950.768695
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.939 − 0.342i)2-s + (1.72 − 0.627i)3-s + (0.766 + 0.642i)4-s + (−0.326 + 1.85i)5-s − 1.83·6-s + (0.598 − 3.39i)7-s + (−0.500 − 0.866i)8-s + (0.278 − 0.233i)9-s + (0.939 − 1.62i)10-s + (1.40 + 2.43i)11-s + (1.72 + 0.627i)12-s + (−2.65 − 2.23i)13-s + (−1.72 + 2.98i)14-s + (0.598 + 3.39i)15-s + (0.173 + 0.984i)16-s + (−2.37 + 1.99i)17-s + ⋯
L(s)  = 1  + (−0.664 − 0.241i)2-s + (0.994 − 0.362i)3-s + (0.383 + 0.321i)4-s + (−0.145 + 0.827i)5-s − 0.748·6-s + (0.226 − 1.28i)7-s + (−0.176 − 0.306i)8-s + (0.0927 − 0.0777i)9-s + (0.297 − 0.514i)10-s + (0.423 + 0.733i)11-s + (0.497 + 0.181i)12-s + (−0.737 − 0.618i)13-s + (−0.460 + 0.797i)14-s + (0.154 + 0.876i)15-s + (0.0434 + 0.246i)16-s + (−0.575 + 0.483i)17-s + ⋯

Functional equation

Λ(s)=(74s/2 ΓC(s) L(s)=((0.928+0.370i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.928 + 0.370i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(74s/2 ΓC(s+1/2) L(s)=((0.928+0.370i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.928 + 0.370i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 7474    =    2⋅372 \cdot 37
Sign: 0.928+0.370i0.928 + 0.370i
Analytic conductor: 0.5908920.590892
Root analytic conductor: 0.7686950.768695
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ74(53,⋅)\chi_{74} (53, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 74, ( :1/2), 0.928+0.370i)(2,\ 74,\ (\ :1/2),\ 0.928 + 0.370i)

Particular Values

L(1)L(1) ≈\approx 0.870674−0.167383i0.870674 - 0.167383i
L(12)L(\frac12) ≈\approx 0.870674−0.167383i0.870674 - 0.167383i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1+(0.939+0.342i)T 1 + (0.939 + 0.342i)T
37 1+(2.56+5.51i)T 1 + (2.56 + 5.51i)T
good3 1+(−1.72+0.627i)T+(2.29−1.92i)T2 1 + (-1.72 + 0.627i)T + (2.29 - 1.92i)T^{2}
5 1+(0.326−1.85i)T+(−4.69−1.71i)T2 1 + (0.326 - 1.85i)T + (-4.69 - 1.71i)T^{2}
7 1+(−0.598+3.39i)T+(−6.57−2.39i)T2 1 + (-0.598 + 3.39i)T + (-6.57 - 2.39i)T^{2}
11 1+(−1.40−2.43i)T+(−5.5+9.52i)T2 1 + (-1.40 - 2.43i)T + (-5.5 + 9.52i)T^{2}
13 1+(2.65+2.23i)T+(2.25+12.8i)T2 1 + (2.65 + 2.23i)T + (2.25 + 12.8i)T^{2}
17 1+(2.37−1.99i)T+(2.95−16.7i)T2 1 + (2.37 - 1.99i)T + (2.95 - 16.7i)T^{2}
19 1+(6.99−2.54i)T+(14.5−12.2i)T2 1 + (6.99 - 2.54i)T + (14.5 - 12.2i)T^{2}
23 1+(−0.321+0.557i)T+(−11.5−19.9i)T2 1 + (-0.321 + 0.557i)T + (-11.5 - 19.9i)T^{2}
29 1+(1.08+1.87i)T+(−14.5+25.1i)T2 1 + (1.08 + 1.87i)T + (-14.5 + 25.1i)T^{2}
31 1−9.90T+31T2 1 - 9.90T + 31T^{2}
41 1+(8.13+6.82i)T+(7.11+40.3i)T2 1 + (8.13 + 6.82i)T + (7.11 + 40.3i)T^{2}
43 1−8.30T+43T2 1 - 8.30T + 43T^{2}
47 1+(−3.92+6.80i)T+(−23.5−40.7i)T2 1 + (-3.92 + 6.80i)T + (-23.5 - 40.7i)T^{2}
53 1+(−0.839−4.76i)T+(−49.8+18.1i)T2 1 + (-0.839 - 4.76i)T + (-49.8 + 18.1i)T^{2}
59 1+(−0.0961−0.545i)T+(−55.4+20.1i)T2 1 + (-0.0961 - 0.545i)T + (-55.4 + 20.1i)T^{2}
61 1+(−5.37−4.50i)T+(10.5+60.0i)T2 1 + (-5.37 - 4.50i)T + (10.5 + 60.0i)T^{2}
67 1+(−0.0366+0.207i)T+(−62.9−22.9i)T2 1 + (-0.0366 + 0.207i)T + (-62.9 - 22.9i)T^{2}
71 1+(4.75−1.72i)T+(54.3−45.6i)T2 1 + (4.75 - 1.72i)T + (54.3 - 45.6i)T^{2}
73 1−10.2T+73T2 1 - 10.2T + 73T^{2}
79 1+(0.330−1.87i)T+(−74.2−27.0i)T2 1 + (0.330 - 1.87i)T + (-74.2 - 27.0i)T^{2}
83 1+(−5.21+4.37i)T+(14.4−81.7i)T2 1 + (-5.21 + 4.37i)T + (14.4 - 81.7i)T^{2}
89 1+(1.68+9.54i)T+(−83.6+30.4i)T2 1 + (1.68 + 9.54i)T + (-83.6 + 30.4i)T^{2}
97 1+(8.52−14.7i)T+(−48.5−84.0i)T2 1 + (8.52 - 14.7i)T + (-48.5 - 84.0i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−14.57163821606939409590157019002, −13.56049440175055657754565301191, −12.37259948542458278860797179981, −10.79591377197980021092610537142, −10.19770910535915263496537469185, −8.659827450074097166176740484543, −7.61627413552245467125509416649, −6.81210004841551863407061843341, −3.98282362841921112276334608742, −2.31994716054009641416243844545, 2.54006356054050893070693495172, 4.71995734231250318682238379607, 6.43193263559365270436746016883, 8.417437348373399069943889553352, 8.724649244667358222488303155745, 9.610186624143061983498738984193, 11.35739359984355807894208009172, 12.35536288540429063027082934257, 13.83495500100852360719255545386, 14.93800715539082527281357025172

Graph of the ZZ-function along the critical line