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Properties

Label 2-234-117.110-c1-0-5
Degree $2$
Conductor $234$
Sign $-0.475 - 0.879i$
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 + (1.60 + 0.649i)3-s + (−0.866 − 0.499i)4-s + (−0.901 + 3.36i)5-s + (−1.04 + 1.38i)6-s + (−0.0396 − 0.0396i)7-s + (0.707 − 0.707i)8-s + (2.15 + 2.08i)9-s + (−3.01 − 1.74i)10-s + (−4.98 − 1.33i)11-s + (−1.06 − 1.36i)12-s + (2.77 − 2.29i)13-s + (0.0485 − 0.0280i)14-s + (−3.63 + 4.81i)15-s + (0.500 + 0.866i)16-s + (1.83 + 3.17i)17-s + ⋯
L(s)  = 1  + (−0.183 + 0.683i)2-s + (0.927 + 0.374i)3-s + (−0.433 − 0.249i)4-s + (−0.403 + 1.50i)5-s + (−0.425 + 0.564i)6-s + (−0.0149 − 0.0149i)7-s + (0.249 − 0.249i)8-s + (0.719 + 0.694i)9-s + (−0.954 − 0.550i)10-s + (−1.50 − 0.402i)11-s + (−0.307 − 0.394i)12-s + (0.770 − 0.637i)13-s + (0.0129 − 0.00749i)14-s + (−0.938 + 1.24i)15-s + (0.125 + 0.216i)16-s + (0.443 + 0.768i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.475 - 0.879i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.475 - 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(234\)    =    \(2 \cdot 3^{2} \cdot 13\)
Sign: $-0.475 - 0.879i$
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: \((2,\ 234,\ (\ :1/2),\ -0.475 - 0.879i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.669505 + 1.12260i\)
\(L(\frac12)\) \(\approx\) \(0.669505 + 1.12260i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.60 - 0.649i)T \)
13 \( 1 + (-2.77 + 2.29i)T \)
good5 \( 1 + (0.901 - 3.36i)T + (-4.33 - 2.5i)T^{2} \)
7 \( 1 + (0.0396 + 0.0396i)T + 7iT^{2} \)
11 \( 1 + (4.98 + 1.33i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (-1.83 - 3.17i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.677 - 0.181i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 - 5.78T + 23T^{2} \)
29 \( 1 + (-2.99 + 1.72i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (5.75 + 1.54i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-7.28 + 1.95i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-2.13 - 2.13i)T + 41iT^{2} \)
43 \( 1 - 5.42iT - 43T^{2} \)
47 \( 1 + (1.29 + 4.82i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + 3.89iT - 53T^{2} \)
59 \( 1 + (0.272 + 1.01i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + 3.00T + 61T^{2} \)
67 \( 1 + (-7.69 + 7.69i)T - 67iT^{2} \)
71 \( 1 + (-3.98 + 14.8i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (-5.86 - 5.86i)T + 73iT^{2} \)
79 \( 1 + (-2.76 + 4.79i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (15.4 - 4.13i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (-4.41 - 16.4i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (1.62 - 1.62i)T - 97iT^{2} \)
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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

−12.92675220388165266903280596158, −10.94938471581652748063940731068, −10.62609371216340196767573754752, −9.576331974384415071785630532115, −8.152910617905726929458370683035, −7.80809505832119627564224826906, −6.65422734885767737278649933223, −5.35981167448514636470074481231, −3.69230829749689899855078097166, −2.77365679724432047358846026384, 1.17191456175733907787720455149, 2.78638290455167849389447593628, 4.22847795921141320193628853571, 5.25376231364198450196080335815, 7.29074419680388161734237902499, 8.167996660519851431544541329924, 8.943583252010918064891533546970, 9.637559262346501582392822301399, 10.95640914107627106397640511996, 12.12397102194215363995348033763

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