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Properties

Label 2-605-5.4-c1-0-11
Degree $2$
Conductor $605$
Sign $0.306 + 0.951i$
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  − 2.52i·2-s − 0.792i·3-s − 4.37·4-s + (0.686 + 2.12i)5-s − 2·6-s + 3.46i·7-s + 5.98i·8-s + 2.37·9-s + (5.37 − 1.73i)10-s + 3.46i·12-s + 8.74·14-s + (1.68 − 0.543i)15-s + 6.37·16-s + 1.58i·17-s − 5.98i·18-s + 4·19-s + ⋯
L(s)  = 1  − 1.78i·2-s − 0.457i·3-s − 2.18·4-s + (0.306 + 0.951i)5-s − 0.816·6-s + 1.30i·7-s + 2.11i·8-s + 0.790·9-s + (1.69 − 0.547i)10-s + 1.00i·12-s + 2.33·14-s + (0.435 − 0.140i)15-s + 1.59·16-s + 0.384i·17-s − 1.41i·18-s + 0.917·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.306 + 0.951i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.306 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.306 + 0.951i$
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: \((2,\ 605,\ (\ :1/2),\ 0.306 + 0.951i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.13937 - 0.829789i\)
\(L(\frac12)\) \(\approx\) \(1.13937 - 0.829789i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.686 - 2.12i)T \)
11 \( 1 \)
good2 \( 1 + 2.52iT - 2T^{2} \)
3 \( 1 + 0.792iT - 3T^{2} \)
7 \( 1 - 3.46iT - 7T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 - 1.58iT - 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 - 0.792iT - 23T^{2} \)
29 \( 1 - 8.74T + 29T^{2} \)
31 \( 1 - 3.37T + 31T^{2} \)
37 \( 1 - 1.08iT - 37T^{2} \)
41 \( 1 + 8.74T + 41T^{2} \)
43 \( 1 + 3.46iT - 43T^{2} \)
47 \( 1 - 6.63iT - 47T^{2} \)
53 \( 1 - 10.0iT - 53T^{2} \)
59 \( 1 - 7.37T + 59T^{2} \)
61 \( 1 - 0.744T + 61T^{2} \)
67 \( 1 + 9.30iT - 67T^{2} \)
71 \( 1 + 10.1T + 71T^{2} \)
73 \( 1 - 6.92iT - 73T^{2} \)
79 \( 1 - 1.25T + 79T^{2} \)
83 \( 1 + 6.63iT - 83T^{2} \)
89 \( 1 + 1.37T + 89T^{2} \)
97 \( 1 + 5.84iT - 97T^{2} \)
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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

−10.43514543590271844527785654902, −9.950783526445393507902468497506, −9.101436849108355547396581663097, −8.135813054304680301963330940786, −6.86155895943070410537506426622, −5.76747495233482591773060691658, −4.57078960261336057308232382301, −3.24076928710729858462834979968, −2.48924736349072704104789548393, −1.46807841895572030158506165825, 0.928349196478761223245163912915, 3.82449024088773432808378203551, 4.68381839900749743337294279008, 5.21174472340115466090675591112, 6.51749051782981139691969967219, 7.18655902884916164388664087066, 8.049329838650943789914981100268, 8.840599013571498797805820709386, 9.865946616985462500816563901458, 10.19983666449200188816903008508

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