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Properties

Label 2-935-935.274-c0-0-3
Degree $2$
Conductor $935$
Sign $-0.856 + 0.516i$
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  + (−0.785 − 0.785i)2-s + 0.234i·4-s + (−0.382 − 0.923i)5-s + (0.636 − 1.53i)7-s + (−0.601 + 0.601i)8-s + (0.707 − 0.707i)9-s + (−0.425 + 1.02i)10-s + (0.923 + 0.382i)11-s − 0.390i·13-s + (−1.70 + 0.707i)14-s + 1.17·16-s + (−0.195 + 0.980i)17-s − 1.11·18-s + (0.216 − 0.0897i)20-s + (−0.425 − 1.02i)22-s + ⋯
L(s)  = 1  + (−0.785 − 0.785i)2-s + 0.234i·4-s + (−0.382 − 0.923i)5-s + (0.636 − 1.53i)7-s + (−0.601 + 0.601i)8-s + (0.707 − 0.707i)9-s + (−0.425 + 1.02i)10-s + (0.923 + 0.382i)11-s − 0.390i·13-s + (−1.70 + 0.707i)14-s + 1.17·16-s + (−0.195 + 0.980i)17-s − 1.11·18-s + (0.216 − 0.0897i)20-s + (−0.425 − 1.02i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.856 + 0.516i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.856 + 0.516i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(935\)    =    \(5 \cdot 11 \cdot 17\)
Sign: $-0.856 + 0.516i$
Analytic conductor: \(0.466625\)
Root analytic conductor: \(0.683100\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{935} (274, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 935,\ (\ :0),\ -0.856 + 0.516i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6895729031\)
\(L(\frac12)\) \(\approx\) \(0.6895729031\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.382 + 0.923i)T \)
11 \( 1 + (-0.923 - 0.382i)T \)
17 \( 1 + (0.195 - 0.980i)T \)
good2 \( 1 + (0.785 + 0.785i)T + iT^{2} \)
3 \( 1 + (-0.707 + 0.707i)T^{2} \)
7 \( 1 + (-0.636 + 1.53i)T + (-0.707 - 0.707i)T^{2} \)
13 \( 1 + 0.390iT - T^{2} \)
19 \( 1 - iT^{2} \)
23 \( 1 + (-0.707 - 0.707i)T^{2} \)
29 \( 1 + (0.707 - 0.707i)T^{2} \)
31 \( 1 + (1.30 - 0.541i)T + (0.707 - 0.707i)T^{2} \)
37 \( 1 + (-0.707 + 0.707i)T^{2} \)
41 \( 1 + (0.707 + 0.707i)T^{2} \)
43 \( 1 + (-0.785 + 0.785i)T - iT^{2} \)
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 \( 1 + (1.70 - 0.707i)T + (0.707 - 0.707i)T^{2} \)
73 \( 1 + (-0.750 - 1.81i)T + (-0.707 + 0.707i)T^{2} \)
79 \( 1 + (-0.707 - 0.707i)T^{2} \)
83 \( 1 + (-1.38 - 1.38i)T + iT^{2} \)
89 \( 1 - 1.84iT - T^{2} \)
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

−9.943069254956621538460434562479, −9.221090939843101247527957077523, −8.467843369271606643043631523289, −7.59910983510297116284038832927, −6.74557097932541583522311690332, −5.42727504224371079824033122913, −4.21925303274830118444759035694, −3.73653734178655387398742685674, −1.66738157359713449631823364755, −0.983746600126137151343462989505, 2.03795268364189739361279878548, 3.23333954218996281369039452611, 4.50816773774931181466610860322, 5.75914277250617379364790650463, 6.53466122112037969737915710357, 7.42171321124978931352097615055, 7.941298166494765859771701252850, 9.025152318875452301865866161768, 9.306867507182092544114691064269, 10.50914053640482386840846106297

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