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Properties

Label 2-234-117.59-c1-0-12
Degree $2$
Conductor $234$
Sign $0.192 + 0.981i$
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 + (1.41 − 1.00i)3-s − 1.00i·4-s + (−0.593 − 2.21i)5-s + (0.290 − 1.70i)6-s + (1.07 + 4.02i)7-s + (−0.707 − 0.707i)8-s + (0.991 − 2.83i)9-s + (−1.98 − 1.14i)10-s + (1.17 + 1.17i)11-s + (−1.00 − 1.41i)12-s + (−3.59 + 0.246i)13-s + (3.60 + 2.08i)14-s + (−3.05 − 2.53i)15-s − 1.00·16-s + (−1.96 − 3.41i)17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (0.815 − 0.578i)3-s − 0.500i·4-s + (−0.265 − 0.990i)5-s + (0.118 − 0.697i)6-s + (0.407 + 1.52i)7-s + (−0.250 − 0.250i)8-s + (0.330 − 0.943i)9-s + (−0.628 − 0.362i)10-s + (0.355 + 0.355i)11-s + (−0.289 − 0.407i)12-s + (−0.997 + 0.0683i)13-s + (0.964 + 0.556i)14-s + (−0.789 − 0.654i)15-s − 0.250·16-s + (−0.477 − 0.827i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.192 + 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.192 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(234\)    =    \(2 \cdot 3^{2} \cdot 13\)
Sign: $0.192 + 0.981i$
Analytic conductor: \(1.86849\)
Root analytic conductor: \(1.36693\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{234} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 234,\ (\ :1/2),\ 0.192 + 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.48743 - 1.22409i\)
\(L(\frac12)\) \(\approx\) \(1.48743 - 1.22409i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.41 + 1.00i)T \)
13 \( 1 + (3.59 - 0.246i)T \)
good5 \( 1 + (0.593 + 2.21i)T + (-4.33 + 2.5i)T^{2} \)
7 \( 1 + (-1.07 - 4.02i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (-1.17 - 1.17i)T + 11iT^{2} \)
17 \( 1 + (1.96 + 3.41i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.878 - 3.27i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-2.53 - 4.39i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 5.23iT - 29T^{2} \)
31 \( 1 + (-2.27 + 0.610i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (-1.21 - 4.55i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (-3.94 - 1.05i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (4.26 + 2.46i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.17 + 4.39i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + 6.69iT - 53T^{2} \)
59 \( 1 + (6.12 + 6.12i)T + 59iT^{2} \)
61 \( 1 + (5.81 - 10.0i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.59 - 13.4i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (-13.5 - 3.63i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (-6.61 + 6.61i)T - 73iT^{2} \)
79 \( 1 + (8.13 + 14.0i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (7.55 + 2.02i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 + (8.95 - 2.40i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-8.27 + 2.21i)T + (84.0 - 48.5i)T^{2} \)
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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

−12.18323991696453275850510130867, −11.56083110448147497375122345972, −9.729970995208698876213799003670, −9.027200580198768691577888475544, −8.260023649423713177638712369559, −6.96899727746681462074552227569, −5.48943758707768533449985622763, −4.53475960988697374356805472083, −2.89401515074157555380167082056, −1.70315998772541220674291992599, 2.72170190372381000859082397901, 3.95196693487184408392351247532, 4.68440177160583202012543887229, 6.57077034630417523960793008406, 7.38600691011094675346125955196, 8.169966456570918081575598800058, 9.483655394906781142931688644604, 10.69935048432052968543005219944, 11.03136736850717133934967308790, 12.67267811402474282257818962057

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