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Properties

Label 2-595-595.489-c0-0-0
Degree $2$
Conductor $595$
Sign $0.992 - 0.122i$
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  − 4-s + (0.707 + 0.707i)5-s + (0.707 − 0.707i)7-s − i·9-s + (−1 + i)11-s + 1.41·13-s + 16-s + (0.707 + 0.707i)17-s + (−0.707 − 0.707i)20-s + 1.00i·25-s + (−0.707 + 0.707i)28-s + (−1 − i)29-s + 1.00·35-s + i·36-s + (1 − i)44-s + (0.707 − 0.707i)45-s + ⋯
L(s)  = 1  − 4-s + (0.707 + 0.707i)5-s + (0.707 − 0.707i)7-s − i·9-s + (−1 + i)11-s + 1.41·13-s + 16-s + (0.707 + 0.707i)17-s + (−0.707 − 0.707i)20-s + 1.00i·25-s + (−0.707 + 0.707i)28-s + (−1 − i)29-s + 1.00·35-s + i·36-s + (1 − i)44-s + (0.707 − 0.707i)45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 595 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.122i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 595 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.122i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(595\)    =    \(5 \cdot 7 \cdot 17\)
Sign: $0.992 - 0.122i$
Analytic conductor: \(0.296943\)
Root analytic conductor: \(0.544925\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{595} (489, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 595,\ (\ :0),\ 0.992 - 0.122i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8938800486\)
\(L(\frac12)\) \(\approx\) \(0.8938800486\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.707 - 0.707i)T \)
7 \( 1 + (-0.707 + 0.707i)T \)
17 \( 1 + (-0.707 - 0.707i)T \)
good2 \( 1 + T^{2} \)
3 \( 1 + iT^{2} \)
11 \( 1 + (1 - i)T - iT^{2} \)
13 \( 1 - 1.41T + T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 + (1 + i)T + iT^{2} \)
31 \( 1 - iT^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + iT^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + 1.41T + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 + iT^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (1 + i)T + iT^{2} \)
73 \( 1 + (1.41 + 1.41i)T + iT^{2} \)
79 \( 1 + (1 - i)T - iT^{2} \)
83 \( 1 - 1.41iT - T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + iT^{2} \)
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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

−10.63646336365504692092290299582, −10.07657196439360392616594610098, −9.311921950930346309971945913659, −8.249281306868908369514893127044, −7.47254041939038334518056080669, −6.27575617279081981496722981098, −5.43520509553983783039974244091, −4.24434555103823740034276703966, −3.36830047096451656233174696508, −1.55508375793091427513291739097, 1.49699539496304710817003797098, 3.08343296203714461743507447948, 4.59855347284396071552998000400, 5.46153615906139493174586345169, 5.75333278769899776754496859040, 7.71161545284990536731656410722, 8.509710481976576844838145717480, 8.832492386135214226746509030325, 9.958210966115801455852850188850, 10.80517933125481302527132041464

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