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Properties

Label 2-605-55.32-c1-0-37
Degree $2$
Conductor $605$
Sign $0.422 + 0.906i$
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  + (−0.738 + 0.738i)2-s + (1.99 − 1.99i)3-s + 0.908i·4-s + (0.742 − 2.10i)5-s + 2.94i·6-s + (−0.388 + 0.388i)7-s + (−2.14 − 2.14i)8-s − 4.93i·9-s + (1.01 + 2.10i)10-s + (1.80 + 1.80i)12-s + (−2.29 − 2.29i)13-s − 0.574i·14-s + (−2.72 − 5.67i)15-s + 1.35·16-s + (4.00 − 4.00i)17-s + (3.64 + 3.64i)18-s + ⋯
L(s)  = 1  + (−0.522 + 0.522i)2-s + (1.14 − 1.14i)3-s + 0.454i·4-s + (0.331 − 0.943i)5-s + 1.20i·6-s + (−0.146 + 0.146i)7-s + (−0.759 − 0.759i)8-s − 1.64i·9-s + (0.319 + 0.666i)10-s + (0.522 + 0.522i)12-s + (−0.636 − 0.636i)13-s − 0.153i·14-s + (−0.702 − 1.46i)15-s + 0.339·16-s + (0.971 − 0.971i)17-s + (0.858 + 0.858i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.422 + 0.906i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.422 + 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.422 + 0.906i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (362, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ 0.422 + 0.906i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.31949 - 0.840832i\)
\(L(\frac12)\) \(\approx\) \(1.31949 - 0.840832i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.742 + 2.10i)T \)
11 \( 1 \)
good2 \( 1 + (0.738 - 0.738i)T - 2iT^{2} \)
3 \( 1 + (-1.99 + 1.99i)T - 3iT^{2} \)
7 \( 1 + (0.388 - 0.388i)T - 7iT^{2} \)
13 \( 1 + (2.29 + 2.29i)T + 13iT^{2} \)
17 \( 1 + (-4.00 + 4.00i)T - 17iT^{2} \)
19 \( 1 - 1.55T + 19T^{2} \)
23 \( 1 + (0.803 - 0.803i)T - 23iT^{2} \)
29 \( 1 - 4.25T + 29T^{2} \)
31 \( 1 + 1.64T + 31T^{2} \)
37 \( 1 + (-0.676 - 0.676i)T + 37iT^{2} \)
41 \( 1 - 8.94iT - 41T^{2} \)
43 \( 1 + (2.55 + 2.55i)T + 43iT^{2} \)
47 \( 1 + (2.87 + 2.87i)T + 47iT^{2} \)
53 \( 1 + (-4.98 + 4.98i)T - 53iT^{2} \)
59 \( 1 - 6.76iT - 59T^{2} \)
61 \( 1 + 9.20iT - 61T^{2} \)
67 \( 1 + (2.62 + 2.62i)T + 67iT^{2} \)
71 \( 1 - 6.85T + 71T^{2} \)
73 \( 1 + (-7.13 - 7.13i)T + 73iT^{2} \)
79 \( 1 + 4.16T + 79T^{2} \)
83 \( 1 + (-8.01 - 8.01i)T + 83iT^{2} \)
89 \( 1 + 3.64iT - 89T^{2} \)
97 \( 1 + (-11.7 - 11.7i)T + 97iT^{2} \)
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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

−9.865332760432092897510080648300, −9.373141706304104886960675117231, −8.475656357516801243655295404402, −7.926978302766921113931944383065, −7.31152754456406233485149146948, −6.33211004411271679442483425036, −5.08406101638046983313401452815, −3.43440775999813151499835026093, −2.50613717840379311651745221681, −0.932631449214235601233510532141, 1.96857232272270873946271031174, 2.93207976586499789665122512635, 3.85800179462073835887980583961, 5.15071129557937268781303945940, 6.27512756856097496120648589219, 7.52368058679721025971083230855, 8.571314921992319731443751723305, 9.330646352716525453615963185925, 10.14386391754014988746529233299, 10.25573140962190551019825810143

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