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Properties

Label 2-595-595.594-c0-0-1
Degree $2$
Conductor $595$
Sign $-0.951 + 0.309i$
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  + 1.17i·2-s + 1.61i·3-s − 0.381·4-s + (−0.951 + 0.309i)5-s − 1.90·6-s − i·7-s + 0.726i·8-s − 1.61·9-s + (−0.363 − 1.11i)10-s − 0.618i·12-s + 1.17·14-s + (−0.500 − 1.53i)15-s − 1.23·16-s + i·17-s − 1.90i·18-s + ⋯
L(s)  = 1  + 1.17i·2-s + 1.61i·3-s − 0.381·4-s + (−0.951 + 0.309i)5-s − 1.90·6-s − i·7-s + 0.726i·8-s − 1.61·9-s + (−0.363 − 1.11i)10-s − 0.618i·12-s + 1.17·14-s + (−0.500 − 1.53i)15-s − 1.23·16-s + i·17-s − 1.90i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 595 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.951 + 0.309i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 595 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.951 + 0.309i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(595\)    =    \(5 \cdot 7 \cdot 17\)
Sign: $-0.951 + 0.309i$
Analytic conductor: \(0.296943\)
Root analytic conductor: \(0.544925\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{595} (594, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 595,\ (\ :0),\ -0.951 + 0.309i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7864964555\)
\(L(\frac12)\) \(\approx\) \(0.7864964555\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.951 - 0.309i)T \)
7 \( 1 + iT \)
17 \( 1 - iT \)
good2 \( 1 - 1.17iT - T^{2} \)
3 \( 1 - 1.61iT - T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - 1.90T + T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 - 1.17T + T^{2} \)
43 \( 1 - 1.17iT - T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + 1.90iT - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + 1.17T + T^{2} \)
67 \( 1 - 1.90iT - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + 1.61iT - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + 0.618iT - T^{2} \)
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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

−11.04656690987089467225795962732, −10.52825059081454728722480902056, −9.647266193066208986050965645542, −8.460653628252996796399924902154, −7.933416430165727194847704014602, −6.89858999818695255675364436534, −5.98596636924254430199104956556, −4.70488887014552577821441399093, −4.20880031637745118566691709463, −3.11460186069803654669947098457, 0.972231739966072897882836686047, 2.31779773827848594370283607151, 3.08777722823703857913177828321, 4.59917460442326889549399134182, 6.00364110454093106044491050771, 6.96689819180773882755835594363, 7.75025967143278718131775532657, 8.642547007087410115357288479410, 9.491927424887615724956285945503, 10.85243996514573187533867565017

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