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Properties

Label 2-605-55.49-c1-0-31
Degree $2$
Conductor $605$
Sign $0.649 + 0.760i$
Analytic cond. $4.83094$
Root an. cond. $2.19794$
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  + (−0.492 + 0.159i)2-s + (1.13 − 1.56i)3-s + (−1.40 + 1.01i)4-s + (−0.809 − 2.08i)5-s + (−0.309 + 0.951i)6-s + (1.96 + 2.70i)7-s + (1.13 − 1.56i)8-s + (−0.226 − 0.696i)9-s + (0.732 + 0.896i)10-s + 3.34i·12-s + (4.03 − 1.31i)13-s + (−1.40 − 1.01i)14-s + (−4.17 − 1.10i)15-s + (0.761 − 2.34i)16-s + (3.67 + 1.19i)17-s + (0.222 + 0.306i)18-s + ⋯
L(s)  = 1  + (−0.348 + 0.113i)2-s + (0.655 − 0.902i)3-s + (−0.700 + 0.509i)4-s + (−0.362 − 0.932i)5-s + (−0.126 + 0.388i)6-s + (0.743 + 1.02i)7-s + (0.401 − 0.552i)8-s + (−0.0754 − 0.232i)9-s + (0.231 + 0.283i)10-s + 0.965i·12-s + (1.11 − 0.363i)13-s + (−0.374 − 0.272i)14-s + (−1.07 − 0.284i)15-s + (0.190 − 0.585i)16-s + (0.891 + 0.289i)17-s + (0.0524 + 0.0722i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.649 + 0.760i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.649 + 0.760i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.649 + 0.760i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ 0.649 + 0.760i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.25601 - 0.579146i\)
\(L(\frac12)\) \(\approx\) \(1.25601 - 0.579146i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.809 + 2.08i)T \)
11 \( 1 \)
good2 \( 1 + (0.492 - 0.159i)T + (1.61 - 1.17i)T^{2} \)
3 \( 1 + (-1.13 + 1.56i)T + (-0.927 - 2.85i)T^{2} \)
7 \( 1 + (-1.96 - 2.70i)T + (-2.16 + 6.65i)T^{2} \)
13 \( 1 + (-4.03 + 1.31i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (-3.67 - 1.19i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (3.39 + 2.46i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 3.48iT - 23T^{2} \)
29 \( 1 + (-5.60 + 4.07i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (2.69 + 8.30i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (1.05 + 1.45i)T + (-11.4 + 35.1i)T^{2} \)
41 \( 1 + (1.40 + 1.01i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 6.45iT - 43T^{2} \)
47 \( 1 + (-6.73 + 9.26i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (-2.06 + 0.671i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (1.02 - 0.745i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (3.62 - 11.1i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 - 2.20iT - 67T^{2} \)
71 \( 1 + (2.53 - 7.79i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (2.87 + 3.96i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (0.452 + 1.39i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-9.41 - 3.05i)T + (67.1 + 48.7i)T^{2} \)
89 \( 1 + 0.464T + 89T^{2} \)
97 \( 1 + (-8.69 + 2.82i)T + (78.4 - 57.0i)T^{2} \)
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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

−10.41776337222339549260708722200, −9.121707327100622993439388067062, −8.517795864782148807919450795412, −8.202658051514142358463810695546, −7.45216205954384585338706851695, −6.00148013549258666635339346877, −4.90658596426343695966820048777, −3.88948344097443069772206753009, −2.40174264273507004082442936626, −1.01289171454161089031837884555, 1.38117254744593153749194629562, 3.35126119116762821613942131724, 4.01378473781363718819451613403, 4.92517295378179579344281352665, 6.30177558006062567498575127258, 7.47657524063313624261280663158, 8.353709376672087023640633726303, 9.048022947614700331410477312421, 10.09804024498232758073806360977, 10.56115125400782047267014034405

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