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Properties

Label 2-234-117.41-c1-0-3
Degree $2$
Conductor $234$
Sign $0.550 - 0.834i$
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  + (−0.965 + 0.258i)2-s + (0.880 + 1.49i)3-s + (0.866 − 0.499i)4-s + (1.93 − 0.517i)5-s + (−1.23 − 1.21i)6-s + (1.87 + 1.87i)7-s + (−0.707 + 0.707i)8-s + (−1.44 + 2.62i)9-s + (−1.73 + 0.999i)10-s + (−1.12 − 4.21i)11-s + (1.50 + 0.851i)12-s + (1.69 − 3.18i)13-s + (−2.29 − 1.32i)14-s + (2.47 + 2.42i)15-s + (0.500 − 0.866i)16-s + (−1.21 + 2.09i)17-s + ⋯
L(s)  = 1  + (−0.683 + 0.183i)2-s + (0.508 + 0.861i)3-s + (0.433 − 0.249i)4-s + (0.863 − 0.231i)5-s + (−0.504 − 0.495i)6-s + (0.709 + 0.709i)7-s + (−0.249 + 0.249i)8-s + (−0.482 + 0.875i)9-s + (−0.547 + 0.316i)10-s + (−0.340 − 1.27i)11-s + (0.435 + 0.245i)12-s + (0.470 − 0.882i)13-s + (−0.614 − 0.354i)14-s + (0.638 + 0.625i)15-s + (0.125 − 0.216i)16-s + (−0.293 + 0.508i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.550 - 0.834i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.550 - 0.834i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(234\)    =    \(2 \cdot 3^{2} \cdot 13\)
Sign: $0.550 - 0.834i$
Analytic conductor: \(1.86849\)
Root analytic conductor: \(1.36693\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{234} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 234,\ (\ :1/2),\ 0.550 - 0.834i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.09605 + 0.590340i\)
\(L(\frac12)\) \(\approx\) \(1.09605 + 0.590340i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 - 0.258i)T \)
3 \( 1 + (-0.880 - 1.49i)T \)
13 \( 1 + (-1.69 + 3.18i)T \)
good5 \( 1 + (-1.93 + 0.517i)T + (4.33 - 2.5i)T^{2} \)
7 \( 1 + (-1.87 - 1.87i)T + 7iT^{2} \)
11 \( 1 + (1.12 + 4.21i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (1.21 - 2.09i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.64 - 6.14i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + 1.22T + 23T^{2} \)
29 \( 1 + (3.87 + 2.23i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.18 - 8.14i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (-2.49 + 9.31i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (4.85 + 4.85i)T + 41iT^{2} \)
43 \( 1 + 10.2iT - 43T^{2} \)
47 \( 1 + (-5.22 - 1.40i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + 0.142iT - 53T^{2} \)
59 \( 1 + (3.88 + 1.03i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + 10.0T + 61T^{2} \)
67 \( 1 + (-1.78 + 1.78i)T - 67iT^{2} \)
71 \( 1 + (-7.11 + 1.90i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (4.15 + 4.15i)T + 73iT^{2} \)
79 \( 1 + (0.161 + 0.280i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.01 - 3.79i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + (-14.2 - 3.83i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (6.20 - 6.20i)T - 97iT^{2} \)
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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

−12.17144078119423542449976571220, −10.87047670898289381622928179796, −10.40798650185799470883882653388, −9.257405934875122971736873049578, −8.535018565610980831006751243938, −7.88747123987019399893579451016, −5.79568561360711879143764641523, −5.47032954551496028519042850845, −3.50874713788117291724269791025, −1.98957473353963113512414796865, 1.52649536209657578202572529442, 2.59284645692781251545966189280, 4.54489396010783974622225322195, 6.34185137391732751163705904837, 7.18506627708218066799612283670, 7.968955856551595116311361922642, 9.240522984711386604644401208749, 9.826794962996286530995221311902, 11.12920115055266462306247378399, 11.83934454387743072145095413194

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