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Properties

Label 2-4608-8.5-c1-0-32
Degree $2$
Conductor $4608$
Sign $-i$
Analytic cond. $36.7950$
Root an. cond. $6.06589$
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  + 4.44i·5-s + 4.87·7-s − 3.46i·11-s − 14.7·25-s + 5.34i·29-s + 10.5·31-s + 21.7i·35-s + 16.7·49-s + 12.4i·53-s + 15.4·55-s + 11.3i·59-s + 9.79·73-s − 16.8i·77-s + 3.32·79-s − 17.3i·83-s + ⋯
L(s)  = 1  + 1.98i·5-s + 1.84·7-s − 1.04i·11-s − 2.95·25-s + 0.993i·29-s + 1.89·31-s + 3.66i·35-s + 2.39·49-s + 1.71i·53-s + 2.07·55-s + 1.47i·59-s + 1.14·73-s − 1.92i·77-s + 0.373·79-s − 1.90i·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4608\)    =    \(2^{9} \cdot 3^{2}\)
Sign: $-i$
Analytic conductor: \(36.7950\)
Root analytic conductor: \(6.06589\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4608} (2305, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4608,\ (\ :1/2),\ -i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.443930346\)
\(L(\frac12)\) \(\approx\) \(2.443930346\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 - 4.44iT - 5T^{2} \)
7 \( 1 - 4.87T + 7T^{2} \)
11 \( 1 + 3.46iT - 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 5.34iT - 29T^{2} \)
31 \( 1 - 10.5T + 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 12.4iT - 53T^{2} \)
59 \( 1 - 11.3iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 9.79T + 73T^{2} \)
79 \( 1 - 3.32T + 79T^{2} \)
83 \( 1 + 17.3iT - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + 2T + 97T^{2} \)
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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

−8.283632503842856099209274300388, −7.75395494495936581025272451927, −7.14128879706872064772708053039, −6.32532514716642567060818660114, −5.72113267740907932269962054296, −4.77562161526074657671257074320, −3.89610146715657798291923493257, −2.99007989615512283408241159818, −2.35490793655286834571939994950, −1.23052105477763498007348545102, 0.73631347300844482772161964543, 1.60543150107650345779559159516, 2.23230526572612488752597462767, 4.03430363287114868279018682855, 4.55814975762840166369673453920, 5.01992451278308531370222713881, 5.58981533536018241561040654705, 6.73631085784849839402364409430, 7.86436488513688297041881714964, 8.128714733570077948338519043057

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