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Properties

Label 2-936-104.19-c1-0-60
Degree $2$
Conductor $936$
Sign $-0.943 - 0.332i$
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  + (−1.13 − 0.841i)2-s + (0.583 + 1.91i)4-s + (−0.00433 − 0.00433i)5-s + (1.57 + 0.421i)7-s + (0.946 − 2.66i)8-s + (0.00127 + 0.00856i)10-s + (−3.23 + 0.865i)11-s + (−3.04 + 1.92i)13-s + (−1.43 − 1.80i)14-s + (−3.31 + 2.23i)16-s + (−4.80 + 2.77i)17-s + (−3.45 − 0.926i)19-s + (0.00575 − 0.0108i)20-s + (4.40 + 1.73i)22-s + (2.76 − 4.78i)23-s + ⋯
L(s)  = 1  + (−0.803 − 0.595i)2-s + (0.291 + 0.956i)4-s + (−0.00193 − 0.00193i)5-s + (0.594 + 0.159i)7-s + (0.334 − 0.942i)8-s + (0.000404 + 0.00270i)10-s + (−0.974 + 0.261i)11-s + (−0.844 + 0.535i)13-s + (−0.383 − 0.482i)14-s + (−0.829 + 0.558i)16-s + (−1.16 + 0.673i)17-s + (−0.793 − 0.212i)19-s + (0.00128 − 0.00241i)20-s + (0.938 + 0.370i)22-s + (0.575 − 0.997i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.943 - 0.332i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.943 - 0.332i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
Sign: $-0.943 - 0.332i$
Analytic conductor: \(7.47399\)
Root analytic conductor: \(2.73386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{936} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 936,\ (\ :1/2),\ -0.943 - 0.332i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00437278 + 0.0255649i\)
\(L(\frac12)\) \(\approx\) \(0.00437278 + 0.0255649i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.13 + 0.841i)T \)
3 \( 1 \)
13 \( 1 + (3.04 - 1.92i)T \)
good5 \( 1 + (0.00433 + 0.00433i)T + 5iT^{2} \)
7 \( 1 + (-1.57 - 0.421i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (3.23 - 0.865i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (4.80 - 2.77i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.45 + 0.926i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-2.76 + 4.78i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.51 - 0.876i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.487 - 0.487i)T + 31iT^{2} \)
37 \( 1 + (3.01 + 11.2i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (1.29 + 4.82i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (9.33 - 5.39i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.87 - 2.87i)T - 47iT^{2} \)
53 \( 1 - 3.71iT - 53T^{2} \)
59 \( 1 + (0.772 - 2.88i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (1.81 - 1.04i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.133 - 0.497i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-0.569 + 2.12i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (1.36 + 1.36i)T + 73iT^{2} \)
79 \( 1 - 12.6iT - 79T^{2} \)
83 \( 1 + (-3.55 + 3.55i)T - 83iT^{2} \)
89 \( 1 + (14.4 - 3.88i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-4.70 - 1.26i)T + (84.0 + 48.5i)T^{2} \)
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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

−9.634529124377948738625555563247, −8.682420987675151978369563290245, −8.230417734720543309189625980668, −7.20102962805138064423965616450, −6.45505368125825282471339454363, −4.91246622959547654705575233608, −4.19834941943955458564635355004, −2.63048585876482900377740420017, −1.96050710550647404626155354509, −0.01467591205926992353214346065, 1.72548363372290765938462086287, 2.96827785310411812386452533879, 4.83098571370728817649394483508, 5.21988867186133095801502353660, 6.48162565397758673752470514108, 7.27964859576254209239197796109, 8.045173671210887448523180249985, 8.674966604526236027011814590875, 9.696057481925381056688134138470, 10.33354872113604112104154870730

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