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Properties

Label 2-936-39.2-c1-0-1
Degree $2$
Conductor $936$
Sign $-0.473 - 0.880i$
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  + (−2.65 − 2.65i)5-s + (1.47 − 0.396i)7-s + (−4.40 − 1.18i)11-s + (−2.78 + 2.29i)13-s + (−2.30 + 3.98i)17-s + (0.611 + 2.28i)19-s + (3.26 + 5.64i)23-s + 9.11i·25-s + (6.53 − 3.77i)29-s + (3.52 − 3.52i)31-s + (−4.98 − 2.87i)35-s + (−1.56 + 5.84i)37-s + (0.0148 − 0.0553i)41-s + (−2.57 − 1.48i)43-s + (−7.10 + 7.10i)47-s + ⋯
L(s)  = 1  + (−1.18 − 1.18i)5-s + (0.559 − 0.149i)7-s + (−1.32 − 0.356i)11-s + (−0.771 + 0.636i)13-s + (−0.557 + 0.966i)17-s + (0.140 + 0.523i)19-s + (0.679 + 1.17i)23-s + 1.82i·25-s + (1.21 − 0.701i)29-s + (0.633 − 0.633i)31-s + (−0.842 − 0.486i)35-s + (−0.257 + 0.961i)37-s + (0.00231 − 0.00863i)41-s + (−0.393 − 0.226i)43-s + (−1.03 + 1.03i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.473 - 0.880i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.473 - 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
Sign: $-0.473 - 0.880i$
Analytic conductor: \(7.47399\)
Root analytic conductor: \(2.73386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{936} (665, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 936,\ (\ :1/2),\ -0.473 - 0.880i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.157350 + 0.263387i\)
\(L(\frac12)\) \(\approx\) \(0.157350 + 0.263387i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.78 - 2.29i)T \)
good5 \( 1 + (2.65 + 2.65i)T + 5iT^{2} \)
7 \( 1 + (-1.47 + 0.396i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (4.40 + 1.18i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (2.30 - 3.98i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.611 - 2.28i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-3.26 - 5.64i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-6.53 + 3.77i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.52 + 3.52i)T - 31iT^{2} \)
37 \( 1 + (1.56 - 5.84i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-0.0148 + 0.0553i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (2.57 + 1.48i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (7.10 - 7.10i)T - 47iT^{2} \)
53 \( 1 - 8.48iT - 53T^{2} \)
59 \( 1 + (2.84 + 10.6i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (0.615 - 1.06i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (8.23 + 2.20i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (12.9 - 3.46i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (-8.01 - 8.01i)T + 73iT^{2} \)
79 \( 1 + 12.4T + 79T^{2} \)
83 \( 1 + (9.98 + 9.98i)T + 83iT^{2} \)
89 \( 1 + (-0.222 - 0.0596i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (2.90 + 10.8i)T + (-84.0 + 48.5i)T^{2} \)
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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

−10.35768323198949693947443475047, −9.417958401605798280823698763450, −8.288856599673766186847239805818, −8.113837014778610545800705016177, −7.23032435774558036168162806614, −5.86510799062719165412684425288, −4.75924735501910634674974808225, −4.42084698872520432842504392794, −3.06658090851243117207296608657, −1.45903954056284834626876222292, 0.14504860070980349765094815645, 2.58063483372013207819488003851, 3.05990166339890862057166331651, 4.60500166754350479594516390358, 5.11613287577145948707939285593, 6.72414805246658522523536246189, 7.23285401997291221884523277376, 7.990637844896354260491271137119, 8.720690316582211631338316124591, 10.15158284881804748568486559937

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