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Properties

Label 2-595-595.104-c0-0-1
Degree $2$
Conductor $595$
Sign $0.905 + 0.425i$
Analytic cond. $0.296943$
Root an. cond. $0.544925$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.30 − 0.541i)3-s + i·4-s + (−0.382 − 0.923i)5-s + (0.382 − 0.923i)7-s + (0.707 − 0.707i)9-s + (−0.707 − 0.292i)11-s + (0.541 + 1.30i)12-s + 1.84i·13-s + (−1 − 0.999i)15-s − 16-s + (−0.382 − 0.923i)17-s + (0.923 − 0.382i)20-s − 1.41i·21-s + (−0.707 + 0.707i)25-s + (0.923 + 0.382i)28-s + (0.707 + 1.70i)29-s + ⋯
L(s)  = 1  + (1.30 − 0.541i)3-s + i·4-s + (−0.382 − 0.923i)5-s + (0.382 − 0.923i)7-s + (0.707 − 0.707i)9-s + (−0.707 − 0.292i)11-s + (0.541 + 1.30i)12-s + 1.84i·13-s + (−1 − 0.999i)15-s − 16-s + (−0.382 − 0.923i)17-s + (0.923 − 0.382i)20-s − 1.41i·21-s + (−0.707 + 0.707i)25-s + (0.923 + 0.382i)28-s + (0.707 + 1.70i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 595 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.905 + 0.425i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 595 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.905 + 0.425i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(595\)    =    \(5 \cdot 7 \cdot 17\)
Sign: $0.905 + 0.425i$
Analytic conductor: \(0.296943\)
Root analytic conductor: \(0.544925\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{595} (104, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 595,\ (\ :0),\ 0.905 + 0.425i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.249394482\)
\(L(\frac12)\) \(\approx\) \(1.249394482\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.382 + 0.923i)T \)
7 \( 1 + (-0.382 + 0.923i)T \)
17 \( 1 + (0.382 + 0.923i)T \)
good2 \( 1 - iT^{2} \)
3 \( 1 + (-1.30 + 0.541i)T + (0.707 - 0.707i)T^{2} \)
11 \( 1 + (0.707 + 0.292i)T + (0.707 + 0.707i)T^{2} \)
13 \( 1 - 1.84iT - T^{2} \)
19 \( 1 - iT^{2} \)
23 \( 1 + (-0.707 - 0.707i)T^{2} \)
29 \( 1 + (-0.707 - 1.70i)T + (-0.707 + 0.707i)T^{2} \)
31 \( 1 + (-0.707 + 0.707i)T^{2} \)
37 \( 1 + (-0.707 + 0.707i)T^{2} \)
41 \( 1 + (0.707 + 0.707i)T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + 0.765iT - T^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 + iT^{2} \)
61 \( 1 + (0.707 + 0.707i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (0.707 - 0.292i)T + (0.707 - 0.707i)T^{2} \)
73 \( 1 + (-0.707 + 0.707i)T^{2} \)
79 \( 1 + (-0.707 - 0.292i)T + (0.707 + 0.707i)T^{2} \)
83 \( 1 + (1.30 + 1.30i)T + iT^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (0.541 + 1.30i)T + (-0.707 + 0.707i)T^{2} \)
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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

−11.04221147520408793392030171703, −9.534675268785707886279601381761, −8.715871933858602457484617661377, −8.359671204860221794478202646773, −7.32117303953840702311405656168, −7.00236454093950306820661350224, −4.86008765082357701053373692412, −4.09034719326582541958196153465, −3.05927409321901959606860819220, −1.76564052711637059397512304399, 2.29619988179659127039867129929, 2.94539823455328708463809651794, 4.26229484295085344182502409849, 5.45924510647367313713146517197, 6.31690999745236698112249655729, 7.84876217994446326952410143934, 8.194462307202249931953909467145, 9.284505598082989149643399363179, 10.20087598429072676245158327982, 10.54250661078813965541913611793

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