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Properties

Label 2-234-117.110-c1-0-8
Degree $2$
Conductor $234$
Sign $0.646 + 0.763i$
Analytic cond. $1.86849$
Root an. cond. $1.36693$
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  + (−0.258 + 0.965i)2-s + (−1.15 + 1.29i)3-s + (−0.866 − 0.499i)4-s + (0.817 − 3.04i)5-s + (−0.947 − 1.44i)6-s + (−2.08 − 2.08i)7-s + (0.707 − 0.707i)8-s + (−0.330 − 2.98i)9-s + (2.73 + 1.57i)10-s + (−0.842 − 0.225i)11-s + (1.64 − 0.539i)12-s + (−0.0715 − 3.60i)13-s + (2.54 − 1.47i)14-s + (2.99 + 4.57i)15-s + (0.500 + 0.866i)16-s + (1.80 + 3.11i)17-s + ⋯
L(s)  = 1  + (−0.183 + 0.683i)2-s + (−0.666 + 0.745i)3-s + (−0.433 − 0.249i)4-s + (0.365 − 1.36i)5-s + (−0.386 − 0.591i)6-s + (−0.786 − 0.786i)7-s + (0.249 − 0.249i)8-s + (−0.110 − 0.993i)9-s + (0.864 + 0.499i)10-s + (−0.254 − 0.0680i)11-s + (0.475 − 0.155i)12-s + (−0.0198 − 0.999i)13-s + (0.681 − 0.393i)14-s + (0.772 + 1.18i)15-s + (0.125 + 0.216i)16-s + (0.436 + 0.756i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.646 + 0.763i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.646 + 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(234\)    =    \(2 \cdot 3^{2} \cdot 13\)
Sign: $0.646 + 0.763i$
Analytic conductor: \(1.86849\)
Root analytic conductor: \(1.36693\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{234} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 234,\ (\ :1/2),\ 0.646 + 0.763i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.599057 - 0.277721i\)
\(L(\frac12)\) \(\approx\) \(0.599057 - 0.277721i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 - 0.965i)T \)
3 \( 1 + (1.15 - 1.29i)T \)
13 \( 1 + (0.0715 + 3.60i)T \)
good5 \( 1 + (-0.817 + 3.04i)T + (-4.33 - 2.5i)T^{2} \)
7 \( 1 + (2.08 + 2.08i)T + 7iT^{2} \)
11 \( 1 + (0.842 + 0.225i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (-1.80 - 3.11i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.90 + 1.31i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 - 3.73T + 23T^{2} \)
29 \( 1 + (-7.44 + 4.29i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.96 + 1.33i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (5.77 - 1.54i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (7.72 + 7.72i)T + 41iT^{2} \)
43 \( 1 + 6.33iT - 43T^{2} \)
47 \( 1 + (-1.63 - 6.10i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 - 6.73iT - 53T^{2} \)
59 \( 1 + (-1.42 - 5.32i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 - 6.36T + 61T^{2} \)
67 \( 1 + (-5.85 + 5.85i)T - 67iT^{2} \)
71 \( 1 + (-1.65 + 6.19i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (-10.6 - 10.6i)T + 73iT^{2} \)
79 \( 1 + (1.30 - 2.26i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-11.0 + 2.96i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (2.10 + 7.84i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (11.9 - 11.9i)T - 97iT^{2} \)
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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

−12.34677667551973034690658898170, −10.64235809335415037471737893923, −10.14144838175629058430742297879, −9.097128861766953144082384357774, −8.294048577081804983738467066579, −6.80337730026939349857853110302, −5.73933516661744511180203917123, −4.92173585474351978694816123815, −3.77965339584959208214003946362, −0.61751188035414591690531273272, 2.10753077327172893559920098240, 3.14533583434654597465246969895, 5.14191632529005620382344376084, 6.49464815641745262480805507677, 6.94568728608342048674683079762, 8.471426248949811056733666615116, 9.695760974095238787466710562327, 10.56098195498160024470776756554, 11.34702760839053301146461651202, 12.20581935246150703566446784062

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