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Properties

Label 2-605-5.4-c1-0-31
Degree $2$
Conductor $605$
Sign $0.774 - 0.632i$
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  + 0.517i·2-s + 1.93i·3-s + 1.73·4-s + (1.73 − 1.41i)5-s − 0.999·6-s − 3.34i·7-s + 1.93i·8-s − 0.732·9-s + (0.732 + 0.896i)10-s + 3.34i·12-s − 4.24i·13-s + 1.73·14-s + (2.73 + 3.34i)15-s + 2.46·16-s + 3.86i·17-s − 0.378i·18-s + ⋯
L(s)  = 1  + 0.366i·2-s + 1.11i·3-s + 0.866·4-s + (0.774 − 0.632i)5-s − 0.408·6-s − 1.26i·7-s + 0.683i·8-s − 0.244·9-s + (0.231 + 0.283i)10-s + 0.965i·12-s − 1.17i·13-s + 0.462·14-s + (0.705 + 0.863i)15-s + 0.616·16-s + 0.937i·17-s − 0.0893i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.774 - 0.632i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.774 - 0.632i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.774 - 0.632i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (364, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ 0.774 - 0.632i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.99430 + 0.710758i\)
\(L(\frac12)\) \(\approx\) \(1.99430 + 0.710758i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.73 + 1.41i)T \)
11 \( 1 \)
good2 \( 1 - 0.517iT - 2T^{2} \)
3 \( 1 - 1.93iT - 3T^{2} \)
7 \( 1 + 3.34iT - 7T^{2} \)
13 \( 1 + 4.24iT - 13T^{2} \)
17 \( 1 - 3.86iT - 17T^{2} \)
19 \( 1 - 4.19T + 19T^{2} \)
23 \( 1 + 3.48iT - 23T^{2} \)
29 \( 1 + 6.92T + 29T^{2} \)
31 \( 1 + 8.73T + 31T^{2} \)
37 \( 1 - 1.79iT - 37T^{2} \)
41 \( 1 - 1.73T + 41T^{2} \)
43 \( 1 - 6.45iT - 43T^{2} \)
47 \( 1 - 11.4iT - 47T^{2} \)
53 \( 1 + 2.17iT - 53T^{2} \)
59 \( 1 - 1.26T + 59T^{2} \)
61 \( 1 + 11.7T + 61T^{2} \)
67 \( 1 - 2.20iT - 67T^{2} \)
71 \( 1 + 8.19T + 71T^{2} \)
73 \( 1 - 4.89iT - 73T^{2} \)
79 \( 1 + 1.46T + 79T^{2} \)
83 \( 1 - 9.89iT - 83T^{2} \)
89 \( 1 + 0.464T + 89T^{2} \)
97 \( 1 + 9.14iT - 97T^{2} \)
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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

−10.61467693339483744527282841534, −10.03470123091481218202022992858, −9.193685654276974073826290647700, −7.989772949590948673344256072888, −7.26483595409943843274859797967, −6.03490858953240965662834417194, −5.30591327477370796445025117828, −4.24313808032640236850281893996, −3.15590357963151962619131020556, −1.43548722526240499851103844497, 1.74402302399033843791325800318, 2.21664224541674993110731154087, 3.37523115874015794447840729163, 5.45130569574316330792314250628, 6.11078711356515679157581910667, 7.17444280522297577118819373214, 7.37262725669464713187426925659, 9.065243711719431692394248357327, 9.579820657118191286954228896550, 10.76569969198394059688773863666

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