Properties

Label 2-608-1.1-c1-0-8
Degree $2$
Conductor $608$
Sign $1$
Analytic cond. $4.85490$
Root an. cond. $2.20338$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.56·3-s + 3.40·5-s − 2.50·7-s − 0.561·9-s + 1.40·11-s + 6.22·13-s + 5.31·15-s − 3.16·17-s − 19-s − 3.91·21-s + 7.91·23-s + 6.60·25-s − 5.56·27-s + 6.22·29-s − 8.13·31-s + 2.19·33-s − 8.52·35-s − 6.13·37-s + 9.71·39-s + 1.68·41-s − 1.71·43-s − 1.91·45-s − 9.84·47-s − 0.729·49-s − 4.94·51-s + 2.78·53-s + 4.78·55-s + ⋯
L(s)  = 1  + 0.901·3-s + 1.52·5-s − 0.946·7-s − 0.187·9-s + 0.423·11-s + 1.72·13-s + 1.37·15-s − 0.767·17-s − 0.229·19-s − 0.853·21-s + 1.64·23-s + 1.32·25-s − 1.07·27-s + 1.15·29-s − 1.46·31-s + 0.382·33-s − 1.44·35-s − 1.00·37-s + 1.55·39-s + 0.263·41-s − 0.261·43-s − 0.285·45-s − 1.43·47-s − 0.104·49-s − 0.691·51-s + 0.382·53-s + 0.645·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(608\)    =    \(2^{5} \cdot 19\)
Sign: $1$
Analytic conductor: \(4.85490\)
Root analytic conductor: \(2.20338\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{608} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 608,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.343932471\)
\(L(\frac12)\) \(\approx\) \(2.343932471\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 + T \)
good3 \( 1 - 1.56T + 3T^{2} \)
5 \( 1 - 3.40T + 5T^{2} \)
7 \( 1 + 2.50T + 7T^{2} \)
11 \( 1 - 1.40T + 11T^{2} \)
13 \( 1 - 6.22T + 13T^{2} \)
17 \( 1 + 3.16T + 17T^{2} \)
23 \( 1 - 7.91T + 23T^{2} \)
29 \( 1 - 6.22T + 29T^{2} \)
31 \( 1 + 8.13T + 31T^{2} \)
37 \( 1 + 6.13T + 37T^{2} \)
41 \( 1 - 1.68T + 41T^{2} \)
43 \( 1 + 1.71T + 43T^{2} \)
47 \( 1 + 9.84T + 47T^{2} \)
53 \( 1 - 2.78T + 53T^{2} \)
59 \( 1 + 9.25T + 59T^{2} \)
61 \( 1 - 6.52T + 61T^{2} \)
67 \( 1 + 1.87T + 67T^{2} \)
71 \( 1 - 13.0T + 71T^{2} \)
73 \( 1 + 6.28T + 73T^{2} \)
79 \( 1 - 11.2T + 79T^{2} \)
83 \( 1 + 1.80T + 83T^{2} \)
89 \( 1 + 7.57T + 89T^{2} \)
97 \( 1 - 4.81T + 97T^{2} \)
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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

−10.54175949275693305108239912440, −9.459273231051144528766447910292, −9.050055381398546220359706494022, −8.393219105454484635380522827517, −6.77389117415901392003830991004, −6.29362316127869186284936383111, −5.28290744138970174470089862419, −3.66468006284354467491265928533, −2.81979441333074951592183675639, −1.59563453040170557506753101078, 1.59563453040170557506753101078, 2.81979441333074951592183675639, 3.66468006284354467491265928533, 5.28290744138970174470089862419, 6.29362316127869186284936383111, 6.77389117415901392003830991004, 8.393219105454484635380522827517, 9.050055381398546220359706494022, 9.459273231051144528766447910292, 10.54175949275693305108239912440

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