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Properties

Label 2-935-935.219-c0-0-1
Degree $2$
Conductor $935$
Sign $0.490 + 0.871i$
Analytic cond. $0.466625$
Root an. cond. $0.683100$
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  + (−1.38 − 1.38i)2-s + 2.84i·4-s + (0.923 − 0.382i)5-s + (0.360 + 0.149i)7-s + (2.56 − 2.56i)8-s + (−0.707 + 0.707i)9-s + (−1.81 − 0.750i)10-s + (0.382 − 0.923i)11-s + 1.11i·13-s + (−0.292 − 0.707i)14-s − 4.26·16-s + (0.555 + 0.831i)17-s + 1.96·18-s + (1.08 + 2.63i)20-s + (−1.81 + 0.750i)22-s + ⋯
L(s)  = 1  + (−1.38 − 1.38i)2-s + 2.84i·4-s + (0.923 − 0.382i)5-s + (0.360 + 0.149i)7-s + (2.56 − 2.56i)8-s + (−0.707 + 0.707i)9-s + (−1.81 − 0.750i)10-s + (0.382 − 0.923i)11-s + 1.11i·13-s + (−0.292 − 0.707i)14-s − 4.26·16-s + (0.555 + 0.831i)17-s + 1.96·18-s + (1.08 + 2.63i)20-s + (−1.81 + 0.750i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.490 + 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.490 + 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(935\)    =    \(5 \cdot 11 \cdot 17\)
Sign: $0.490 + 0.871i$
Analytic conductor: \(0.466625\)
Root analytic conductor: \(0.683100\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{935} (219, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 935,\ (\ :0),\ 0.490 + 0.871i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6086396678\)
\(L(\frac12)\) \(\approx\) \(0.6086396678\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.923 + 0.382i)T \)
11 \( 1 + (-0.382 + 0.923i)T \)
17 \( 1 + (-0.555 - 0.831i)T \)
good2 \( 1 + (1.38 + 1.38i)T + iT^{2} \)
3 \( 1 + (0.707 - 0.707i)T^{2} \)
7 \( 1 + (-0.360 - 0.149i)T + (0.707 + 0.707i)T^{2} \)
13 \( 1 - 1.11iT - T^{2} \)
19 \( 1 - iT^{2} \)
23 \( 1 + (0.707 + 0.707i)T^{2} \)
29 \( 1 + (-0.707 + 0.707i)T^{2} \)
31 \( 1 + (-0.541 - 1.30i)T + (-0.707 + 0.707i)T^{2} \)
37 \( 1 + (0.707 - 0.707i)T^{2} \)
41 \( 1 + (-0.707 - 0.707i)T^{2} \)
43 \( 1 + (-1.38 + 1.38i)T - iT^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 - iT^{2} \)
61 \( 1 + (-0.707 - 0.707i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (0.292 + 0.707i)T + (-0.707 + 0.707i)T^{2} \)
73 \( 1 + (1.53 - 0.636i)T + (0.707 - 0.707i)T^{2} \)
79 \( 1 + (0.707 + 0.707i)T^{2} \)
83 \( 1 + (1.17 + 1.17i)T + iT^{2} \)
89 \( 1 - 0.765iT - T^{2} \)
97 \( 1 + (-0.707 + 0.707i)T^{2} \)
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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

−10.23530166645743660499992328436, −9.241860430964522943855078930976, −8.679735541818531479195861159218, −8.234814983159168321712145958185, −7.04113043545332046341575235829, −5.83978905419958886770682596970, −4.51928999795569269700635037713, −3.29998205839509601134539024334, −2.22263899775100979882562705132, −1.37121072053558722098846936822, 1.14575928188779777798592660633, 2.61286609232304785891014338220, 4.73210420028312741097929108031, 5.73803863861162818177143388219, 6.18408026094386064446494696167, 7.19389100583287515766191311517, 7.78475056234902994117809630916, 8.739427773813838266082954343845, 9.606369275592433185977942922599, 9.855841043791329683118304247420

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