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Properties

Label 2-605-11.4-c1-0-25
Degree $2$
Conductor $605$
Sign $0.394 + 0.918i$
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  + (0.809 + 0.587i)2-s + (−0.927 + 2.85i)3-s + (−0.309 − 0.951i)4-s + (−0.809 + 0.587i)5-s + (−2.42 + 1.76i)6-s + (−0.927 − 2.85i)7-s + (0.927 − 2.85i)8-s + (−4.85 − 3.52i)9-s − 10-s + 2.99·12-s + (−3.23 − 2.35i)13-s + (0.927 − 2.85i)14-s + (−0.927 − 2.85i)15-s + (0.809 − 0.587i)16-s + (−1.85 − 5.70i)18-s + (1.23 − 3.80i)19-s + ⋯
L(s)  = 1  + (0.572 + 0.415i)2-s + (−0.535 + 1.64i)3-s + (−0.154 − 0.475i)4-s + (−0.361 + 0.262i)5-s + (−0.990 + 0.719i)6-s + (−0.350 − 1.07i)7-s + (0.327 − 1.00i)8-s + (−1.61 − 1.17i)9-s − 0.316·10-s + 0.866·12-s + (−0.897 − 0.652i)13-s + (0.247 − 0.762i)14-s + (−0.239 − 0.736i)15-s + (0.202 − 0.146i)16-s + (−0.437 − 1.34i)18-s + (0.283 − 0.872i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.394 + 0.918i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.394 + 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.394 + 0.918i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ 0.394 + 0.918i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.489485 - 0.322488i\)
\(L(\frac12)\) \(\approx\) \(0.489485 - 0.322488i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.809 - 0.587i)T \)
11 \( 1 \)
good2 \( 1 + (-0.809 - 0.587i)T + (0.618 + 1.90i)T^{2} \)
3 \( 1 + (0.927 - 2.85i)T + (-2.42 - 1.76i)T^{2} \)
7 \( 1 + (0.927 + 2.85i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (3.23 + 2.35i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-1.23 + 3.80i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 8T + 23T^{2} \)
29 \( 1 + (-1.85 - 5.70i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-1.61 - 1.17i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (2.47 + 7.60i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (1.54 - 4.75i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 5T + 43T^{2} \)
47 \( 1 + (0.927 - 2.85i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (3.23 + 2.35i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (0.618 + 1.90i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-8.89 + 6.46i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 13T + 67T^{2} \)
71 \( 1 + (1.61 - 1.17i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (2.47 + 7.60i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (8.09 + 5.87i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (3.23 - 2.35i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - T + 89T^{2} \)
97 \( 1 + (-6.47 - 4.70i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.35689185411108091536179782578, −9.961173516342668371135079802551, −9.107940321674159098617775878489, −7.61880610988925005955761734586, −6.65398580025625962532903583065, −5.65317208821220037234345754557, −4.79965826237156350001113941975, −4.15487183417568456685304694854, −3.26935150372612190660382129586, −0.27669842401542515469935513239, 1.89075250441913589942215105599, 2.72756225843641035421928372655, 4.23339119669243075436047087455, 5.46670749696114166845279095066, 6.18534576004986685578290837291, 7.32442127702582878885075089311, 8.026959781035782677278021620372, 8.743704462335276058463400454628, 10.05123928848904119682016680120, 11.62123335775150903612487557058

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