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Properties

Label 2-605-5.4-c1-0-35
Degree $2$
Conductor $605$
Sign $-1$
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  − 2.49i·2-s − 4.23·4-s + 2.23·5-s − 3.08i·7-s + 5.58i·8-s + 3·9-s − 5.58i·10-s + 1.90i·13-s − 7.70·14-s + 5.47·16-s − 8.08i·17-s − 7.49i·18-s − 9.47·20-s + 5.00·25-s + 4.76·26-s + ⋯
L(s)  = 1  − 1.76i·2-s − 2.11·4-s + 0.999·5-s − 1.16i·7-s + 1.97i·8-s + 9-s − 1.76i·10-s + 0.529i·13-s − 2.06·14-s + 1.36·16-s − 1.95i·17-s − 1.76i·18-s − 2.11·20-s + 1.00·25-s + 0.934·26-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-1$
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: \((2,\ 605,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.56116i\)
\(L(\frac12)\) \(\approx\) \(1.56116i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - 2.23T \)
11 \( 1 \)
good2 \( 1 + 2.49iT - 2T^{2} \)
3 \( 1 - 3T^{2} \)
7 \( 1 + 3.08iT - 7T^{2} \)
13 \( 1 - 1.90iT - 13T^{2} \)
17 \( 1 + 8.08iT - 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + 8.94T + 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 13.0iT - 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 4T + 59T^{2} \)
61 \( 1 + 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 + 11.8iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 0.728iT - 83T^{2} \)
89 \( 1 - 13.4T + 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−10.35193795676495664043977280175, −9.451050358269076952320925644088, −9.298686373497790958905779558510, −7.58493797868114238595442546196, −6.69245220103188484435876803412, −5.05998464086956486955126877504, −4.34959750042247961506898204308, −3.23112208064544733888896798367, −2.00658937323569381458871732384, −0.953751842303099005902313268726, 1.93086453900231116528494668270, 3.87882076135721165959321579650, 5.19503777080631978638760048124, 5.75836689030499676939960622157, 6.47523948457421079185322869497, 7.40588690468398587926988855619, 8.446275951636704845128452023227, 9.005940317800539511251787689309, 9.884299416894840787038080325797, 10.72598114755842887378441383380

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