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Properties

Label 2-605-55.9-c1-0-19
Degree $2$
Conductor $605$
Sign $-0.261 - 0.965i$
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  + (2.40 + 0.780i)2-s + (0.465 + 0.640i)3-s + (3.53 + 2.56i)4-s + (−1.81 + 1.31i)5-s + (0.618 + 1.90i)6-s + (−2.03 + 2.80i)7-s + (3.51 + 4.84i)8-s + (0.733 − 2.25i)9-s + (−5.37 + 1.73i)10-s + 3.46i·12-s + (−7.07 + 5.13i)14-s + (−1.68 − 0.551i)15-s + (1.96 + 6.06i)16-s + (1.50 − 0.489i)17-s + (3.51 − 4.84i)18-s + (3.23 − 2.35i)19-s + ⋯
L(s)  = 1  + (1.69 + 0.551i)2-s + (0.268 + 0.370i)3-s + (1.76 + 1.28i)4-s + (−0.810 + 0.585i)5-s + (0.252 + 0.776i)6-s + (−0.769 + 1.05i)7-s + (1.24 + 1.71i)8-s + (0.244 − 0.752i)9-s + (−1.69 + 0.547i)10-s + 0.999i·12-s + (−1.89 + 1.37i)14-s + (−0.434 − 0.142i)15-s + (0.492 + 1.51i)16-s + (0.365 − 0.118i)17-s + (0.829 − 1.14i)18-s + (0.742 − 0.539i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.261 - 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.261 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.261 - 0.965i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ -0.261 - 0.965i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.14092 + 2.79744i\)
\(L(\frac12)\) \(\approx\) \(2.14092 + 2.79744i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.81 - 1.31i)T \)
11 \( 1 \)
good2 \( 1 + (-2.40 - 0.780i)T + (1.61 + 1.17i)T^{2} \)
3 \( 1 + (-0.465 - 0.640i)T + (-0.927 + 2.85i)T^{2} \)
7 \( 1 + (2.03 - 2.80i)T + (-2.16 - 6.65i)T^{2} \)
13 \( 1 + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (-1.50 + 0.489i)T + (13.7 - 9.99i)T^{2} \)
19 \( 1 + (-3.23 + 2.35i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 0.792iT - 23T^{2} \)
29 \( 1 + (-7.07 - 5.13i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-1.04 + 3.20i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-0.639 + 0.879i)T + (-11.4 - 35.1i)T^{2} \)
41 \( 1 + (7.07 - 5.13i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 3.46iT - 43T^{2} \)
47 \( 1 + (3.89 + 5.36i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (-9.60 - 3.12i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (5.96 + 4.33i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (0.230 + 0.708i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 9.30iT - 67T^{2} \)
71 \( 1 + (3.12 + 9.62i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (4.07 - 5.60i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (0.387 - 1.19i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (6.30 - 2.04i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 + 1.37T + 89T^{2} \)
97 \( 1 + (5.55 + 1.80i)T + (78.4 + 57.0i)T^{2} \)
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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

−11.38931889965353922261571102065, −10.11346613307327007304280435191, −9.086471079489090554086576605398, −7.982326353238616116826051599344, −6.87930723384522119365016337059, −6.40016434397471498404052047528, −5.32019269735733243058643877022, −4.31940955153488867206417683165, −3.24533061041368427347553979600, −2.89789926880006571231282173367, 1.26943620081642641107012654383, 2.85995587464993519616192503057, 3.80014507967035321441414934911, 4.53449794468490096526965420322, 5.48855862520701456276508917670, 6.72905149089759880013438947031, 7.42871447884344897556740835553, 8.426075197998643696335800650639, 10.04386388399712358658980690679, 10.55109471180905494904734682167

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