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Properties

Label 2-936-9.4-c1-0-25
Degree $2$
Conductor $936$
Sign $0.704 + 0.709i$
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  + (−0.377 + 1.69i)3-s + (2.15 − 3.73i)5-s + (−0.803 − 1.39i)7-s + (−2.71 − 1.27i)9-s + (1.64 + 2.85i)11-s + (−0.5 + 0.866i)13-s + (5.49 + 5.05i)15-s + 1.73·17-s + 0.746·19-s + (2.65 − 0.832i)21-s + (1.46 − 2.53i)23-s + (−6.78 − 11.7i)25-s + (3.18 − 4.10i)27-s + (−3.78 − 6.55i)29-s + (3.86 − 6.68i)31-s + ⋯
L(s)  = 1  + (−0.218 + 0.975i)3-s + (0.963 − 1.66i)5-s + (−0.303 − 0.526i)7-s + (−0.904 − 0.425i)9-s + (0.497 + 0.861i)11-s + (−0.138 + 0.240i)13-s + (1.41 + 1.30i)15-s + 0.421·17-s + 0.171·19-s + (0.579 − 0.181i)21-s + (0.304 − 0.528i)23-s + (−1.35 − 2.35i)25-s + (0.613 − 0.789i)27-s + (−0.703 − 1.21i)29-s + (0.693 − 1.20i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.704 + 0.709i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.704 + 0.709i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
Sign: $0.704 + 0.709i$
Analytic conductor: \(7.47399\)
Root analytic conductor: \(2.73386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{936} (625, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 936,\ (\ :1/2),\ 0.704 + 0.709i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.46007 - 0.607921i\)
\(L(\frac12)\) \(\approx\) \(1.46007 - 0.607921i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.377 - 1.69i)T \)
13 \( 1 + (0.5 - 0.866i)T \)
good5 \( 1 + (-2.15 + 3.73i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (0.803 + 1.39i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.64 - 2.85i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 - 1.73T + 17T^{2} \)
19 \( 1 - 0.746T + 19T^{2} \)
23 \( 1 + (-1.46 + 2.53i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.78 + 6.55i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.86 + 6.68i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 2.38T + 37T^{2} \)
41 \( 1 + (-3.90 + 6.75i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.51 + 4.34i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-6.69 - 11.5i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 3.49T + 53T^{2} \)
59 \( 1 + (2.47 - 4.29i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.48 - 7.76i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.83 + 4.90i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 12.2T + 71T^{2} \)
73 \( 1 + 5.51T + 73T^{2} \)
79 \( 1 + (0.231 + 0.401i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.27 + 5.68i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 7.69T + 89T^{2} \)
97 \( 1 + (0.310 + 0.538i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.803190658506611508188906369305, −9.315232791519436300732095565899, −8.655398287177604748790797299712, −7.50929873434450099128727350965, −6.15012333819954442945826130581, −5.54406670472481641278822772865, −4.48087789917561621135816896643, −4.11760600802421073081944456599, −2.31371383791637254011388624791, −0.810075118324892615525740047403, 1.51757268575472643208112800706, 2.77021369597831143412165117098, 3.28557424823698067346393539186, 5.45650859239679895629994559341, 5.95582121431060811598583138833, 6.76388859171759742243470895196, 7.29942347219265410046605907143, 8.447345039304676374778369763938, 9.385441591168809253648753948844, 10.26583465433657818035318540553

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