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Properties

Label 2-935-935.769-c0-0-0
Degree $2$
Conductor $935$
Sign $-0.266 - 0.963i$
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  + 0.765i·2-s + 0.414·4-s + (−0.707 + 0.707i)5-s + (0.541 + 0.541i)7-s + 1.08i·8-s − i·9-s + (−0.541 − 0.541i)10-s + (0.707 + 0.707i)11-s − 1.84·13-s + (−0.414 + 0.414i)14-s − 0.414·16-s + (0.923 + 0.382i)17-s + 0.765·18-s + (−0.292 + 0.292i)20-s + (−0.541 + 0.541i)22-s + ⋯
L(s)  = 1  + 0.765i·2-s + 0.414·4-s + (−0.707 + 0.707i)5-s + (0.541 + 0.541i)7-s + 1.08i·8-s − i·9-s + (−0.541 − 0.541i)10-s + (0.707 + 0.707i)11-s − 1.84·13-s + (−0.414 + 0.414i)14-s − 0.414·16-s + (0.923 + 0.382i)17-s + 0.765·18-s + (−0.292 + 0.292i)20-s + (−0.541 + 0.541i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(935\)    =    \(5 \cdot 11 \cdot 17\)
Sign: $-0.266 - 0.963i$
Analytic conductor: \(0.466625\)
Root analytic conductor: \(0.683100\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{935} (769, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 935,\ (\ :0),\ -0.266 - 0.963i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.069409914\)
\(L(\frac12)\) \(\approx\) \(1.069409914\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.707 - 0.707i)T \)
11 \( 1 + (-0.707 - 0.707i)T \)
17 \( 1 + (-0.923 - 0.382i)T \)
good2 \( 1 - 0.765iT - T^{2} \)
3 \( 1 + iT^{2} \)
7 \( 1 + (-0.541 - 0.541i)T + iT^{2} \)
13 \( 1 + 1.84T + T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 - iT^{2} \)
29 \( 1 + iT^{2} \)
31 \( 1 - iT^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 - iT^{2} \)
43 \( 1 + 0.765iT - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - 2iT - T^{2} \)
61 \( 1 - iT^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (-1 + i)T - iT^{2} \)
73 \( 1 + (-1.30 + 1.30i)T - iT^{2} \)
79 \( 1 - iT^{2} \)
83 \( 1 + 1.84iT - T^{2} \)
89 \( 1 + 1.41T + T^{2} \)
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

−10.45364605035024158183122956515, −9.641344688166194467277314587609, −8.642367960082306806368245792466, −7.66393152393969041813776535563, −7.18963685409293831444754646076, −6.44661231964220470043817708328, −5.46586508719889092285714206017, −4.42949317607835257575459025414, −3.17695818807701487844462389225, −2.06962350107495292629393826455, 1.12840326653314114791936579917, 2.42147691403431254215949640905, 3.59325952840474112936563738462, 4.59424684778081612882746867730, 5.36412715375762428210706129085, 6.85902578342442687683289848831, 7.62764006064965429435830873791, 8.171515287649138350167097766293, 9.474870831757365019189025890150, 10.06478353330752669355307597272

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