Properties

Label 2-722-361.138-c1-0-28
Degree $2$
Conductor $722$
Sign $-0.999 + 0.0347i$
Analytic cond. $5.76519$
Root an. cond. $2.40108$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.870 − 0.492i)2-s + (−1.27 − 1.19i)3-s + (0.515 − 0.856i)4-s + (−0.338 − 0.0124i)5-s + (−1.69 − 0.413i)6-s + (−0.521 − 2.66i)7-s + (0.0275 − 0.999i)8-s + (0.00323 + 0.0502i)9-s + (−0.300 + 0.155i)10-s + (−0.404 − 0.550i)11-s + (−1.68 + 0.475i)12-s + (0.641 + 0.274i)13-s + (−1.76 − 2.06i)14-s + (0.416 + 0.420i)15-s + (−0.467 − 0.883i)16-s + (0.862 − 0.0158i)17-s + ⋯
L(s)  = 1  + (0.615 − 0.347i)2-s + (−0.735 − 0.689i)3-s + (0.257 − 0.428i)4-s + (−0.151 − 0.00556i)5-s + (−0.692 − 0.168i)6-s + (−0.197 − 1.00i)7-s + (0.00974 − 0.353i)8-s + (0.00107 + 0.0167i)9-s + (−0.0951 + 0.0492i)10-s + (−0.121 − 0.165i)11-s + (−0.485 + 0.137i)12-s + (0.178 + 0.0761i)13-s + (−0.472 − 0.552i)14-s + (0.107 + 0.108i)15-s + (−0.116 − 0.220i)16-s + (0.209 − 0.00384i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0347i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0347i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(722\)    =    \(2 \cdot 19^{2}\)
Sign: $-0.999 + 0.0347i$
Analytic conductor: \(5.76519\)
Root analytic conductor: \(2.40108\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{722} (499, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 722,\ (\ :1/2),\ -0.999 + 0.0347i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0194553 - 1.12069i\)
\(L(\frac12)\) \(\approx\) \(0.0194553 - 1.12069i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.870 + 0.492i)T \)
19 \( 1 + (0.0718 - 4.35i)T \)
good3 \( 1 + (1.27 + 1.19i)T + (0.192 + 2.99i)T^{2} \)
5 \( 1 + (0.338 + 0.0124i)T + (4.98 + 0.367i)T^{2} \)
7 \( 1 + (0.521 + 2.66i)T + (-6.48 + 2.63i)T^{2} \)
11 \( 1 + (0.404 + 0.550i)T + (-3.28 + 10.4i)T^{2} \)
13 \( 1 + (-0.641 - 0.274i)T + (8.97 + 9.40i)T^{2} \)
17 \( 1 + (-0.862 + 0.0158i)T + (16.9 - 0.624i)T^{2} \)
23 \( 1 + (3.97 + 1.20i)T + (19.1 + 12.7i)T^{2} \)
29 \( 1 + (2.89 + 2.33i)T + (6.08 + 28.3i)T^{2} \)
31 \( 1 + (2.57 + 6.87i)T + (-23.3 + 20.3i)T^{2} \)
37 \( 1 + (-0.0746 + 0.170i)T + (-25.0 - 27.2i)T^{2} \)
41 \( 1 + (-7.81 - 0.719i)T + (40.3 + 7.49i)T^{2} \)
43 \( 1 + (0.312 + 6.80i)T + (-42.8 + 3.94i)T^{2} \)
47 \( 1 + (2.23 - 2.34i)T + (-2.15 - 46.9i)T^{2} \)
53 \( 1 + (-3.86 - 13.1i)T + (-44.6 + 28.5i)T^{2} \)
59 \( 1 + (5.04 + 10.9i)T + (-38.3 + 44.8i)T^{2} \)
61 \( 1 + (-3.40 + 0.697i)T + (56.0 - 23.9i)T^{2} \)
67 \( 1 + (-3.77 + 0.558i)T + (64.1 - 19.4i)T^{2} \)
71 \( 1 + (7.60 + 1.55i)T + (65.2 + 27.9i)T^{2} \)
73 \( 1 + (1.94 - 3.51i)T + (-38.7 - 61.8i)T^{2} \)
79 \( 1 + (-11.3 + 7.23i)T + (33.0 - 71.7i)T^{2} \)
83 \( 1 + (-1.48 + 10.7i)T + (-79.8 - 22.5i)T^{2} \)
89 \( 1 + (1.36 + 2.46i)T + (-47.3 + 75.3i)T^{2} \)
97 \( 1 + (-0.986 - 0.145i)T + (92.8 + 28.1i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.23856457001197752411445837535, −9.383382283880844028168873487910, −7.87397246458412040908916474309, −7.29204586928014563597513843969, −6.14231856471287791770460157749, −5.76720368251040350737671240019, −4.26177087363140322808987733228, −3.58455080634202197060923357685, −1.89153612442468600931194957691, −0.50304807977678944822988052722, 2.29006229634214362365410703864, 3.58982519367096376077360092015, 4.67174196084917981300990865826, 5.46070056894668640426264664313, 6.05368922616042060893299920425, 7.19415651319834754498607030505, 8.196338166900504119911283501356, 9.169723905303024891814327262137, 10.04526422823735354019354612789, 11.03251702134139931352396963426

Graph of the $Z$-function along the critical line