Properties

Label 2-605-121.16-c1-0-6
Degree $2$
Conductor $605$
Sign $0.936 - 0.349i$
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  + (−1.80 − 0.206i)2-s + (−1.29 − 0.938i)3-s + (1.25 + 0.291i)4-s + (−0.564 + 0.825i)5-s + (2.13 + 1.95i)6-s + (−0.0685 − 0.0704i)7-s + (1.22 + 0.435i)8-s + (−0.139 − 0.430i)9-s + (1.18 − 1.36i)10-s + (−0.376 − 3.29i)11-s + (−1.34 − 1.55i)12-s + (−2.20 + 3.66i)13-s + (0.108 + 0.141i)14-s + (1.50 − 0.536i)15-s + (−4.41 − 2.17i)16-s + (−3.22 + 2.63i)17-s + ⋯
L(s)  = 1  + (−1.27 − 0.146i)2-s + (−0.745 − 0.541i)3-s + (0.625 + 0.145i)4-s + (−0.252 + 0.369i)5-s + (0.870 + 0.798i)6-s + (−0.0258 − 0.0266i)7-s + (0.431 + 0.153i)8-s + (−0.0466 − 0.143i)9-s + (0.375 − 0.433i)10-s + (−0.113 − 0.993i)11-s + (−0.387 − 0.447i)12-s + (−0.612 + 1.01i)13-s + (0.0290 + 0.0377i)14-s + (0.388 − 0.138i)15-s + (−1.10 − 0.542i)16-s + (−0.782 + 0.639i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.936 - 0.349i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.936 - 0.349i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.936 - 0.349i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ 0.936 - 0.349i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.356174 + 0.0642786i\)
\(L(\frac12)\) \(\approx\) \(0.356174 + 0.0642786i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.564 - 0.825i)T \)
11 \( 1 + (0.376 + 3.29i)T \)
good2 \( 1 + (1.80 + 0.206i)T + (1.94 + 0.452i)T^{2} \)
3 \( 1 + (1.29 + 0.938i)T + (0.927 + 2.85i)T^{2} \)
7 \( 1 + (0.0685 + 0.0704i)T + (-0.199 + 6.99i)T^{2} \)
13 \( 1 + (2.20 - 3.66i)T + (-6.06 - 11.4i)T^{2} \)
17 \( 1 + (3.22 - 2.63i)T + (3.37 - 16.6i)T^{2} \)
19 \( 1 + (0.263 + 0.0150i)T + (18.8 + 2.16i)T^{2} \)
23 \( 1 + (-0.379 + 2.64i)T + (-22.0 - 6.47i)T^{2} \)
29 \( 1 + (0.892 - 2.29i)T + (-21.3 - 19.6i)T^{2} \)
31 \( 1 + (-5.76 - 2.43i)T + (21.6 + 22.2i)T^{2} \)
37 \( 1 + (0.824 - 9.60i)T + (-36.4 - 6.30i)T^{2} \)
41 \( 1 + (-5.23 + 2.95i)T + (21.1 - 35.1i)T^{2} \)
43 \( 1 + (-4.86 + 1.42i)T + (36.1 - 23.2i)T^{2} \)
47 \( 1 + (-2.13 + 10.5i)T + (-43.2 - 18.2i)T^{2} \)
53 \( 1 + (-6.05 + 2.97i)T + (32.3 - 41.9i)T^{2} \)
59 \( 1 + (-4.92 - 2.77i)T + (30.4 + 50.5i)T^{2} \)
61 \( 1 + (-7.14 + 0.819i)T + (59.4 - 13.8i)T^{2} \)
67 \( 1 + (6.40 - 14.0i)T + (-43.8 - 50.6i)T^{2} \)
71 \( 1 + (-2.93 - 11.1i)T + (-61.8 + 34.9i)T^{2} \)
73 \( 1 + (-4.39 - 8.33i)T + (-41.2 + 60.2i)T^{2} \)
79 \( 1 + (-0.439 - 15.3i)T + (-78.8 + 4.51i)T^{2} \)
83 \( 1 + (-14.7 - 2.54i)T + (78.1 + 27.8i)T^{2} \)
89 \( 1 + (8.67 + 5.57i)T + (36.9 + 80.9i)T^{2} \)
97 \( 1 + (-4.94 - 7.23i)T + (-35.1 + 90.3i)T^{2} \)
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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

−10.66707150517758915685016656275, −9.910350089868705231183197109449, −8.781665327792236330322442707614, −8.355624296159524976529772421762, −6.99064602721341410491655448330, −6.69175092552802299730366734881, −5.39901236696133855797729731299, −4.05020379801935807576861641827, −2.37655667999973907388542376992, −0.883994965465621572936736766774, 0.47569530126626548410980752092, 2.33597877226171935750180602198, 4.33027504419812437752761459059, 4.96153161557235255000241988281, 6.14647369263205383394858201995, 7.56749170290461224714885916883, 7.77451871547030139340293563929, 9.144598743482442541589963167631, 9.621194410050701589608409388083, 10.52187162356021034792500019769

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