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Properties

Label 2-936-13.9-c1-0-1
Degree $2$
Conductor $936$
Sign $-0.406 - 0.913i$
Analytic cond. $7.47399$
Root an. cond. $2.73386$
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  − 0.678·5-s + (−0.160 − 0.278i)7-s + (−2.10 + 3.65i)11-s + (−2.28 − 2.78i)13-s + (1.76 + 3.06i)17-s + (2.10 + 3.65i)19-s + (−0.787 + 1.36i)23-s − 4.53·25-s + (−4.55 + 7.89i)29-s + 2.53·31-s + (0.109 + 0.188i)35-s + (0.0175 − 0.0303i)37-s + (−1.12 + 1.95i)41-s + (3.37 + 5.85i)43-s − 5.57·47-s + ⋯
L(s)  = 1  − 0.303·5-s + (−0.0607 − 0.105i)7-s + (−0.635 + 1.10i)11-s + (−0.634 − 0.772i)13-s + (0.429 + 0.743i)17-s + (0.483 + 0.838i)19-s + (−0.164 + 0.284i)23-s − 0.907·25-s + (−0.846 + 1.46i)29-s + 0.456·31-s + (0.0184 + 0.0319i)35-s + (0.00288 − 0.00499i)37-s + (−0.175 + 0.304i)41-s + (0.515 + 0.892i)43-s − 0.813·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.406 - 0.913i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.406 - 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
Sign: $-0.406 - 0.913i$
Analytic conductor: \(7.47399\)
Root analytic conductor: \(2.73386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{936} (217, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 936,\ (\ :1/2),\ -0.406 - 0.913i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.453273 + 0.698114i\)
\(L(\frac12)\) \(\approx\) \(0.453273 + 0.698114i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
13 \( 1 + (2.28 + 2.78i)T \)
good5 \( 1 + 0.678T + 5T^{2} \)
7 \( 1 + (0.160 + 0.278i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.10 - 3.65i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1.76 - 3.06i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.10 - 3.65i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.787 - 1.36i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.55 - 7.89i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 2.53T + 31T^{2} \)
37 \( 1 + (-0.0175 + 0.0303i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.12 - 1.95i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.37 - 5.85i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 5.57T + 47T^{2} \)
53 \( 1 + 4.67T + 53T^{2} \)
59 \( 1 + (-4.86 - 8.42i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.96 + 5.13i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.41 - 11.1i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-4.78 - 8.29i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 6.07T + 73T^{2} \)
79 \( 1 - 4.61T + 79T^{2} \)
83 \( 1 + 10.9T + 83T^{2} \)
89 \( 1 + (2 - 3.46i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.94 + 5.10i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.02468921750897213226898855822, −9.877957212312692523945181371867, −8.510789710198369306516934091284, −7.69066754188397852718192812771, −7.21020069800876697642296604161, −5.88610601254053390732483677958, −5.12428513109502511531692792234, −4.04777711705201069131632371100, −3.00151211257813740737618810884, −1.65030309417744172783924293998, 0.37981173124376751830964265212, 2.28432816658805819803791747962, 3.32342004773158713377004769434, 4.47198010119208306927467564107, 5.42853341579840920350056730167, 6.31888514369639979607512170558, 7.42129307474540474511439161870, 7.989907123332332146607987413331, 9.068381679592105364161473617464, 9.680335358394015971167410870514

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