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Properties

Label 2-605-55.8-c1-0-23
Degree $2$
Conductor $605$
Sign $0.937 + 0.348i$
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  + (−0.482 − 0.0763i)2-s + (0.517 − 1.01i)3-s + (−1.67 − 0.544i)4-s + (2.05 + 0.885i)5-s + (−0.327 + 0.450i)6-s + (2.59 − 1.32i)7-s + (1.63 + 0.833i)8-s + (1.00 + 1.37i)9-s + (−0.922 − 0.583i)10-s + (−1.41 + 1.41i)12-s + (−0.457 + 2.89i)13-s + (−1.35 + 0.440i)14-s + (1.96 − 1.62i)15-s + (2.12 + 1.54i)16-s + (0.824 + 5.20i)17-s + (−0.377 − 0.740i)18-s + ⋯
L(s)  = 1  + (−0.341 − 0.0540i)2-s + (0.298 − 0.586i)3-s + (−0.837 − 0.272i)4-s + (0.918 + 0.395i)5-s + (−0.133 + 0.183i)6-s + (0.982 − 0.500i)7-s + (0.578 + 0.294i)8-s + (0.333 + 0.458i)9-s + (−0.291 − 0.184i)10-s + (−0.409 + 0.409i)12-s + (−0.126 + 0.801i)13-s + (−0.362 + 0.117i)14-s + (0.506 − 0.419i)15-s + (0.531 + 0.385i)16-s + (0.199 + 1.26i)17-s + (−0.0889 − 0.174i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.937 + 0.348i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.937 + 0.348i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.937 + 0.348i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (118, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ 0.937 + 0.348i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.50644 - 0.271122i\)
\(L(\frac12)\) \(\approx\) \(1.50644 - 0.271122i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-2.05 - 0.885i)T \)
11 \( 1 \)
good2 \( 1 + (0.482 + 0.0763i)T + (1.90 + 0.618i)T^{2} \)
3 \( 1 + (-0.517 + 1.01i)T + (-1.76 - 2.42i)T^{2} \)
7 \( 1 + (-2.59 + 1.32i)T + (4.11 - 5.66i)T^{2} \)
13 \( 1 + (0.457 - 2.89i)T + (-12.3 - 4.01i)T^{2} \)
17 \( 1 + (-0.824 - 5.20i)T + (-16.1 + 5.25i)T^{2} \)
19 \( 1 + (1.26 + 3.89i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (2.12 + 2.12i)T + 23iT^{2} \)
29 \( 1 + (0.817 - 2.51i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-5.45 + 3.96i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (0.528 + 1.03i)T + (-21.7 + 29.9i)T^{2} \)
41 \( 1 + (-3.38 + 1.09i)T + (33.1 - 24.0i)T^{2} \)
43 \( 1 + (-5.07 + 5.07i)T - 43iT^{2} \)
47 \( 1 + (3.28 + 1.67i)T + (27.6 + 38.0i)T^{2} \)
53 \( 1 + (1.45 + 0.231i)T + (50.4 + 16.3i)T^{2} \)
59 \( 1 + (1.52 + 0.496i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (-4.07 + 5.61i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 + (-1.31 + 1.31i)T - 67iT^{2} \)
71 \( 1 + (2.32 + 1.68i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (6.71 + 13.1i)T + (-42.9 + 59.0i)T^{2} \)
79 \( 1 + (12.3 - 8.96i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (6.60 - 1.04i)T + (78.9 - 25.6i)T^{2} \)
89 \( 1 - 11.1iT - 89T^{2} \)
97 \( 1 + (1.57 - 9.94i)T + (-92.2 - 29.9i)T^{2} \)
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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

−10.54431290164947155121023642916, −9.763570790190628500381573516647, −8.786631605510814012128762486436, −8.076137247620557659352517659454, −7.18481183915087699085976805782, −6.15274613237730410226681699198, −4.96464338169967930549773225448, −4.16540095562814628474831021100, −2.22361947507726259666242956149, −1.37432759714823209563609927076, 1.24056959787188387795083227194, 2.90813404827498649251885133940, 4.30484369639649473335374577961, 5.02642949240418976681797458828, 5.92071389861906527934060038497, 7.45692268100624110822128004916, 8.380115731433450769121792928557, 8.950187859559319353285440910732, 9.919509711405572474794358255463, 10.10039595617640360310690563251

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