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Properties

Label 2-234-117.22-c1-0-13
Degree $2$
Conductor $234$
Sign $0.782 + 0.622i$
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  + 2-s + (1.25 − 1.19i)3-s + 4-s + (1.44 − 2.49i)5-s + (1.25 − 1.19i)6-s + (−2.39 + 4.13i)7-s + 8-s + (0.126 − 2.99i)9-s + (1.44 − 2.49i)10-s − 3.77·11-s + (1.25 − 1.19i)12-s + (−0.0450 + 3.60i)13-s + (−2.39 + 4.13i)14-s + (−1.18 − 4.84i)15-s + 16-s + (1.17 + 2.04i)17-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.721 − 0.692i)3-s + 0.5·4-s + (0.644 − 1.11i)5-s + (0.510 − 0.489i)6-s + (−0.903 + 1.56i)7-s + 0.353·8-s + (0.0422 − 0.999i)9-s + (0.455 − 0.789i)10-s − 1.13·11-s + (0.360 − 0.346i)12-s + (−0.0124 + 0.999i)13-s + (−0.638 + 1.10i)14-s + (−0.307 − 1.25i)15-s + 0.250·16-s + (0.286 + 0.495i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.782 + 0.622i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.782 + 0.622i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(234\)    =    \(2 \cdot 3^{2} \cdot 13\)
Sign: $0.782 + 0.622i$
Analytic conductor: \(1.86849\)
Root analytic conductor: \(1.36693\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{234} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 234,\ (\ :1/2),\ 0.782 + 0.622i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.05625 - 0.718113i\)
\(L(\frac12)\) \(\approx\) \(2.05625 - 0.718113i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + (-1.25 + 1.19i)T \)
13 \( 1 + (0.0450 - 3.60i)T \)
good5 \( 1 + (-1.44 + 2.49i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (2.39 - 4.13i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + 3.77T + 11T^{2} \)
17 \( 1 + (-1.17 - 2.04i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.90 + 5.03i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.39 - 4.15i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 6.93T + 29T^{2} \)
31 \( 1 + (-0.501 + 0.868i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.590 + 1.02i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.693 + 1.20i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.98 - 6.90i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (5.24 + 9.09i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 3.79T + 53T^{2} \)
59 \( 1 - 1.10T + 59T^{2} \)
61 \( 1 + (-1.37 + 2.37i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.93 - 6.81i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (2.34 + 4.06i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 10.6T + 73T^{2} \)
79 \( 1 + (3.65 + 6.32i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-6.00 - 10.4i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-2.23 + 3.86i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.82 - 3.16i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.46074361933752981978016753356, −11.62050813340558966290163218604, −9.841268256461033376997279694220, −9.007796175503652781971508843194, −8.339479725528770029205647653131, −6.78736550582475158079605650437, −5.87293491358721076879421123219, −4.86721810986340006555749364100, −3.01243579233279822622495647044, −1.99846298924497798100186045719, 2.73907915704608242834992455106, 3.41030089291624410874792719364, 4.74321772575256634769948030560, 6.16075180087439881955094304462, 7.19342242803428299737115842461, 8.122016451175204637210839500732, 10.00440839636336520683916507594, 10.30696077374042724139078922891, 10.81450534617615501171031734762, 12.71880180279112029319855205557

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