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Properties

Label 2-234-117.11-c1-0-0
Degree $2$
Conductor $234$
Sign $0.704 - 0.709i$
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  + (−0.707 + 0.707i)2-s + (−1.69 − 0.356i)3-s − 1.00i·4-s + (−3.71 − 0.995i)5-s + (1.45 − 0.946i)6-s + (2.11 + 0.566i)7-s + (0.707 + 0.707i)8-s + (2.74 + 1.20i)9-s + (3.33 − 1.92i)10-s + (2.96 + 2.96i)11-s + (−0.356 + 1.69i)12-s + (3.17 + 1.70i)13-s + (−1.89 + 1.09i)14-s + (5.94 + 3.00i)15-s − 1.00·16-s + (0.465 − 0.806i)17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (−0.978 − 0.205i)3-s − 0.500i·4-s + (−1.66 − 0.445i)5-s + (0.592 − 0.386i)6-s + (0.798 + 0.214i)7-s + (0.250 + 0.250i)8-s + (0.915 + 0.402i)9-s + (1.05 − 0.607i)10-s + (0.892 + 0.892i)11-s + (−0.102 + 0.489i)12-s + (0.880 + 0.473i)13-s + (−0.506 + 0.292i)14-s + (1.53 + 0.777i)15-s − 0.250·16-s + (0.112 − 0.195i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.704 - 0.709i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.704 - 0.709i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(234\)    =    \(2 \cdot 3^{2} \cdot 13\)
Sign: $0.704 - 0.709i$
Analytic conductor: \(1.86849\)
Root analytic conductor: \(1.36693\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{234} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 234,\ (\ :1/2),\ 0.704 - 0.709i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.554161 + 0.230782i\)
\(L(\frac12)\) \(\approx\) \(0.554161 + 0.230782i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
3 \( 1 + (1.69 + 0.356i)T \)
13 \( 1 + (-3.17 - 1.70i)T \)
good5 \( 1 + (3.71 + 0.995i)T + (4.33 + 2.5i)T^{2} \)
7 \( 1 + (-2.11 - 0.566i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (-2.96 - 2.96i)T + 11iT^{2} \)
17 \( 1 + (-0.465 + 0.806i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.86 - 0.501i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-3.91 + 6.78i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 7.40iT - 29T^{2} \)
31 \( 1 + (0.333 - 1.24i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (-4.93 - 1.32i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-2.83 - 10.5i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-1.70 + 0.984i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-9.40 + 2.52i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 - 2.04iT - 53T^{2} \)
59 \( 1 + (4.46 + 4.46i)T + 59iT^{2} \)
61 \( 1 + (0.430 + 0.745i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.23 - 0.865i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-2.90 - 10.8i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (2.45 - 2.45i)T - 73iT^{2} \)
79 \( 1 + (-6.38 + 11.0i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.45 + 5.43i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 + (-0.522 + 1.95i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (4.55 - 17.0i)T + (-84.0 - 48.5i)T^{2} \)
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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

−12.07578998148154646789788989661, −11.35448341683894467127814937497, −10.68656528702851839675689342785, −9.102706514636723040329597019705, −8.251313077538800319487731922614, −7.29074223056606425613505012454, −6.47173348881359645085253576011, −4.88176992001524119399789837493, −4.22733521714075668291283585591, −1.19951222237010550292316409826, 0.868719491280413783597944810222, 3.57281658370701463493349188941, 4.28127848474147422436442164770, 5.94718790309612117201941503289, 7.27687359549650393725888161413, 8.043232303912022482491985705902, 9.139481520783847895941535917405, 10.67524389357872015855876694510, 11.19680634781020834959460306793, 11.58360264022619616107063731330

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