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Properties

Label 2-234-13.7-c2-0-6
Degree $2$
Conductor $234$
Sign $0.926 + 0.376i$
Analytic cond. $6.37603$
Root an. cond. $2.52508$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.36 − 0.366i)2-s + (1.73 − i)4-s + (−0.323 − 0.323i)5-s + (7.67 + 2.05i)7-s + (1.99 − 2i)8-s + (−0.560 − 0.323i)10-s + (−1.44 − 5.40i)11-s + (12.8 + 1.93i)13-s + 11.2·14-s + (1.99 − 3.46i)16-s + (1.74 − 1.00i)17-s + (1.19 − 4.47i)19-s + (−0.884 − 0.237i)20-s + (−3.95 − 6.85i)22-s + (15.0 + 8.71i)23-s + ⋯
L(s)  = 1  + (0.683 − 0.183i)2-s + (0.433 − 0.250i)4-s + (−0.0647 − 0.0647i)5-s + (1.09 + 0.293i)7-s + (0.249 − 0.250i)8-s + (−0.0560 − 0.0323i)10-s + (−0.131 − 0.491i)11-s + (0.988 + 0.148i)13-s + 0.803·14-s + (0.124 − 0.216i)16-s + (0.102 − 0.0591i)17-s + (0.0631 − 0.235i)19-s + (−0.0442 − 0.0118i)20-s + (−0.179 − 0.311i)22-s + (0.656 + 0.378i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.926 + 0.376i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.926 + 0.376i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(234\)    =    \(2 \cdot 3^{2} \cdot 13\)
Sign: $0.926 + 0.376i$
Analytic conductor: \(6.37603\)
Root analytic conductor: \(2.52508\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{234} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 234,\ (\ :1),\ 0.926 + 0.376i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.53528 - 0.494825i\)
\(L(\frac12)\) \(\approx\) \(2.53528 - 0.494825i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.36 + 0.366i)T \)
3 \( 1 \)
13 \( 1 + (-12.8 - 1.93i)T \)
good5 \( 1 + (0.323 + 0.323i)T + 25iT^{2} \)
7 \( 1 + (-7.67 - 2.05i)T + (42.4 + 24.5i)T^{2} \)
11 \( 1 + (1.44 + 5.40i)T + (-104. + 60.5i)T^{2} \)
17 \( 1 + (-1.74 + 1.00i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-1.19 + 4.47i)T + (-312. - 180.5i)T^{2} \)
23 \( 1 + (-15.0 - 8.71i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (8.25 - 14.2i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (9.27 + 9.27i)T + 961iT^{2} \)
37 \( 1 + (-9.12 - 34.0i)T + (-1.18e3 + 684.5i)T^{2} \)
41 \( 1 + (58.6 - 15.7i)T + (1.45e3 - 840.5i)T^{2} \)
43 \( 1 + (45.2 - 26.1i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (8.73 - 8.73i)T - 2.20e3iT^{2} \)
53 \( 1 - 23.4T + 2.80e3T^{2} \)
59 \( 1 + (34.0 + 9.13i)T + (3.01e3 + 1.74e3i)T^{2} \)
61 \( 1 + (13.9 + 24.1i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (33.0 - 8.85i)T + (3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + (19.7 - 73.8i)T + (-4.36e3 - 2.52e3i)T^{2} \)
73 \( 1 + (18.9 - 18.9i)T - 5.32e3iT^{2} \)
79 \( 1 + 142.T + 6.24e3T^{2} \)
83 \( 1 + (-91.7 - 91.7i)T + 6.88e3iT^{2} \)
89 \( 1 + (40.4 + 150. i)T + (-6.85e3 + 3.96e3i)T^{2} \)
97 \( 1 + (-29.1 + 108. i)T + (-8.14e3 - 4.70e3i)T^{2} \)
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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

−11.66436990775034886801744674226, −11.30140560049163388461537493681, −10.22570829625930972581714977946, −8.801457207132927797513752099093, −7.996811268052659052948262741414, −6.63523193435569298220980136067, −5.49999482618331360292693113166, −4.53440570495795154307271166818, −3.15985769050763886223758436922, −1.51229838257516350067464423770, 1.69243411689925572477843296005, 3.48402338941726289953468181328, 4.67048754110292123164034259134, 5.65531150665835039258955749784, 6.97828669635375591229193486430, 7.88072291084428443985045474011, 8.890939285788622382017303477212, 10.40022483592185274702268583925, 11.16418980688193106391028944340, 11.99983944957916436256092742923

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