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Properties

Label 2-935-935.934-c0-0-5
Degree $2$
Conductor $935$
Sign $1$
Analytic cond. $0.466625$
Root an. cond. $0.683100$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.80·2-s − 0.445·3-s + 2.24·4-s + 5-s + 0.801·6-s − 2.24·8-s − 0.801·9-s − 1.80·10-s + 11-s − 12-s + 1.24·13-s − 0.445·15-s + 1.80·16-s + 17-s + 1.44·18-s + 2.24·20-s − 1.80·22-s − 1.80·23-s + 1.00·24-s + 25-s − 2.24·26-s + 0.801·27-s − 1.80·29-s + 0.801·30-s − 1.00·32-s − 0.445·33-s − 1.80·34-s + ⋯
L(s)  = 1  − 1.80·2-s − 0.445·3-s + 2.24·4-s + 5-s + 0.801·6-s − 2.24·8-s − 0.801·9-s − 1.80·10-s + 11-s − 12-s + 1.24·13-s − 0.445·15-s + 1.80·16-s + 17-s + 1.44·18-s + 2.24·20-s − 1.80·22-s − 1.80·23-s + 1.00·24-s + 25-s − 2.24·26-s + 0.801·27-s − 1.80·29-s + 0.801·30-s − 1.00·32-s − 0.445·33-s − 1.80·34-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(935\)    =    \(5 \cdot 11 \cdot 17\)
Sign: $1$
Analytic conductor: \(0.466625\)
Root analytic conductor: \(0.683100\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{935} (934, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 935,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4835460024\)
\(L(\frac12)\) \(\approx\) \(0.4835460024\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - T \)
11 \( 1 - T \)
17 \( 1 - T \)
good2 \( 1 + 1.80T + T^{2} \)
3 \( 1 + 0.445T + T^{2} \)
7 \( 1 - T^{2} \)
13 \( 1 - 1.24T + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + 1.80T + T^{2} \)
29 \( 1 + 1.80T + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - 1.24T + T^{2} \)
41 \( 1 + 0.445T + T^{2} \)
43 \( 1 + 0.445T + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 - 1.24T + T^{2} \)
61 \( 1 - 1.24T + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + 0.445T + T^{2} \)
83 \( 1 - 1.24T + T^{2} \)
89 \( 1 + 0.445T + T^{2} \)
97 \( 1 + 1.80T + T^{2} \)
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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

−10.01812638170962073534924338090, −9.492638722702879543517543876874, −8.693149487086164377518195223791, −8.075798673019834902801182938552, −6.94437212916747104243304006184, −6.05548424389580179399099228133, −5.70207217003271100973742802262, −3.66884682526385728531712783340, −2.22300392954886064082123647102, −1.17739137584930038965141713979, 1.17739137584930038965141713979, 2.22300392954886064082123647102, 3.66884682526385728531712783340, 5.70207217003271100973742802262, 6.05548424389580179399099228133, 6.94437212916747104243304006184, 8.075798673019834902801182938552, 8.693149487086164377518195223791, 9.492638722702879543517543876874, 10.01812638170962073534924338090

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