Properties

Label 2-462-77.13-c1-0-8
Degree $2$
Conductor $462$
Sign $0.971 - 0.237i$
Analytic cond. $3.68908$
Root an. cond. $1.92070$
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  + (0.951 + 0.309i)2-s + (−0.587 − 0.809i)3-s + (0.809 + 0.587i)4-s + (2.13 − 0.693i)5-s + (−0.309 − 0.951i)6-s + (0.112 + 2.64i)7-s + (0.587 + 0.809i)8-s + (−0.309 + 0.951i)9-s + 2.24·10-s + (1.68 + 2.85i)11-s − 0.999i·12-s + (0.720 − 2.21i)13-s + (−0.709 + 2.54i)14-s + (−1.81 − 1.31i)15-s + (0.309 + 0.951i)16-s + (−0.988 − 3.04i)17-s + ⋯
L(s)  = 1  + (0.672 + 0.218i)2-s + (−0.339 − 0.467i)3-s + (0.404 + 0.293i)4-s + (0.953 − 0.309i)5-s + (−0.126 − 0.388i)6-s + (0.0424 + 0.999i)7-s + (0.207 + 0.286i)8-s + (−0.103 + 0.317i)9-s + 0.709·10-s + (0.508 + 0.861i)11-s − 0.288i·12-s + (0.199 − 0.614i)13-s + (−0.189 + 0.681i)14-s + (−0.468 − 0.340i)15-s + (0.0772 + 0.237i)16-s + (−0.239 − 0.738i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 - 0.237i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.971 - 0.237i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.971 - 0.237i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ 0.971 - 0.237i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.19003 + 0.263574i\)
\(L(\frac12)\) \(\approx\) \(2.19003 + 0.263574i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.951 - 0.309i)T \)
3 \( 1 + (0.587 + 0.809i)T \)
7 \( 1 + (-0.112 - 2.64i)T \)
11 \( 1 + (-1.68 - 2.85i)T \)
good5 \( 1 + (-2.13 + 0.693i)T + (4.04 - 2.93i)T^{2} \)
13 \( 1 + (-0.720 + 2.21i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (0.988 + 3.04i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (0.0237 - 0.0172i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 3.27T + 23T^{2} \)
29 \( 1 + (-4.01 + 5.52i)T + (-8.96 - 27.5i)T^{2} \)
31 \( 1 + (-1.88 - 0.612i)T + (25.0 + 18.2i)T^{2} \)
37 \( 1 + (2.68 + 1.94i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (8.83 - 6.42i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 7.48iT - 43T^{2} \)
47 \( 1 + (0.343 + 0.473i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (2.39 - 7.37i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (5.23 - 7.21i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (4.44 + 13.6i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 - 5.48T + 67T^{2} \)
71 \( 1 + (-3.72 - 11.4i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-0.0950 - 0.0690i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (15.6 + 5.07i)T + (63.9 + 46.4i)T^{2} \)
83 \( 1 + (2.28 + 7.02i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 11.1iT - 89T^{2} \)
97 \( 1 + (12.3 + 4.00i)T + (78.4 + 57.0i)T^{2} \)
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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

−11.42696900191405839304409146106, −10.16798191488273631305148662158, −9.290122896297000433746465237527, −8.325266064155150813482506868143, −7.11974753871573877359365880304, −6.24154210907160680954545059498, −5.46294116156050015427435371215, −4.66144438269889896198980234296, −2.86396243863036570058996465191, −1.73898709751441499770703299381, 1.47989247593168317567430273195, 3.18714525100798563236263289823, 4.17133702959405384729663302033, 5.24056598706376145501271527072, 6.34324340057349752717120789291, 6.81645612816466465920756854930, 8.423074702897772076217376934996, 9.508102296012673144379820608092, 10.39456426868408473179212805160, 10.91170148304977316944847535932

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