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Properties

Label 2-595-595.594-c0-0-4
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

\(L(\frac{1}{2})\) \(\approx\) \(1.079183070\)
\(L(\frac12)\) \(\approx\) \(1.079183070\)
\(L(1)\) 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

−10.69254699021251203521391104744, −10.24115080454413778197823415044, −9.557983764540068994329382297687, −8.828136484826744138048124775365, −7.23911702360785415414265546147, −6.09061589318524147059314282134, −4.94485676776141944379751063346, −3.88479226252338813219282300852, −3.29685507612423088611875851878, −1.85398317211527482765458946232, 1.79079941191697797368039226207, 2.65436942874941726731857832039, 5.21121777077990595460145030814, 5.66273778471854458366942396908, 6.60592445154722630961686978006, 7.10385884786149803335364930751, 8.157870554683301188656748107855, 8.756999416078930940103537200235, 9.672010026487032435835779286403, 11.27153191203733090183373992814

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