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Properties

Label 2-595-595.489-c0-0-1
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

\(L(\frac{1}{2})\) \(\approx\) \(0.09032397746\)
\(L(\frac12)\) \(\approx\) \(0.09032397746\)
\(L(1)\) 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

−10.09743667132571647421381158248, −9.443791279120376988523075579503, −8.980446335066052443626422811552, −7.86380657804579272836053808470, −7.06785484517721604186883057229, −5.61586758707162161618097458945, −4.82827045801829154003294777363, −3.96391880066902145342679056734, −2.59895230610057306180734589747, −0.10143899253577293048531083236, 2.68968995654678575470923767835, 3.75157907312926211754391531390, 4.72743234054808983161556392147, 5.73042299282190757867389871636, 7.11184448652881892255316077113, 7.75394880620284699888510025152, 8.565092854939187223711381514395, 9.693987795208842029183281565348, 10.61059461722508123820687954518, 10.87483096896470957053017443072

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