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Properties

Label 2-935-17.16-c1-0-20
Degree $2$
Conductor $935$
Sign $0.910 - 0.414i$
Analytic cond. $7.46601$
Root an. cond. $2.73240$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.06·2-s − 0.620i·3-s + 2.25·4-s + i·5-s + 1.28i·6-s + 0.540i·7-s − 0.526·8-s + 2.61·9-s − 2.06i·10-s + i·11-s − 1.40i·12-s + 2.88·13-s − 1.11i·14-s + 0.620·15-s − 3.42·16-s + (3.75 − 1.70i)17-s + ⋯
L(s)  = 1  − 1.45·2-s − 0.358i·3-s + 1.12·4-s + 0.447i·5-s + 0.522i·6-s + 0.204i·7-s − 0.185·8-s + 0.871·9-s − 0.652i·10-s + 0.301i·11-s − 0.404i·12-s + 0.799·13-s − 0.298i·14-s + 0.160·15-s − 0.856·16-s + (0.910 − 0.414i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 - 0.414i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 935 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.910 - 0.414i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(935\)    =    \(5 \cdot 11 \cdot 17\)
Sign: $0.910 - 0.414i$
Analytic conductor: \(7.46601\)
Root analytic conductor: \(2.73240\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{935} (441, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 935,\ (\ :1/2),\ 0.910 - 0.414i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.797457 + 0.172994i\)
\(L(\frac12)\) \(\approx\) \(0.797457 + 0.172994i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - iT \)
11 \( 1 - iT \)
17 \( 1 + (-3.75 + 1.70i)T \)
good2 \( 1 + 2.06T + 2T^{2} \)
3 \( 1 + 0.620iT - 3T^{2} \)
7 \( 1 - 0.540iT - 7T^{2} \)
13 \( 1 - 2.88T + 13T^{2} \)
19 \( 1 + 2.80T + 19T^{2} \)
23 \( 1 + 0.435iT - 23T^{2} \)
29 \( 1 - 7.32iT - 29T^{2} \)
31 \( 1 - 5.49iT - 31T^{2} \)
37 \( 1 + 9.54iT - 37T^{2} \)
41 \( 1 - 7.38iT - 41T^{2} \)
43 \( 1 + 11.8T + 43T^{2} \)
47 \( 1 - 6.31T + 47T^{2} \)
53 \( 1 - 0.779T + 53T^{2} \)
59 \( 1 - 2.87T + 59T^{2} \)
61 \( 1 - 3.04iT - 61T^{2} \)
67 \( 1 - 0.961T + 67T^{2} \)
71 \( 1 + 7.83iT - 71T^{2} \)
73 \( 1 - 2.54iT - 73T^{2} \)
79 \( 1 - 4.42iT - 79T^{2} \)
83 \( 1 - 5.98T + 83T^{2} \)
89 \( 1 - 11.2T + 89T^{2} \)
97 \( 1 + 10.6iT - 97T^{2} \)
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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

−10.19758876806655341609256518493, −9.194945599740285916732506689003, −8.557364107587865868939953459740, −7.60396656377919627068563646879, −7.07036569087649707711502685981, −6.25445615423785193064135166062, −4.89190752519349757413463928726, −3.56151063382996749099095095547, −2.11978321362895293985080932461, −1.09513209428287068270307875969, 0.812259984657833795996830308750, 1.92542066717225073002207237283, 3.68416161711546605782622811060, 4.57533664499561575959983912234, 5.87853772675179957481296951072, 6.86934181436637331365543539364, 7.83788810102403334258937553480, 8.397800771873361563634589381682, 9.194995063519349407939141471969, 10.03672329001735723070192418548

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