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Properties

Label 2-936-104.61-c1-0-54
Degree $2$
Conductor $936$
Sign $0.435 + 0.900i$
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  + (1.41 + 0.0601i)2-s + (1.99 + 0.170i)4-s − 0.556i·5-s + (−2.30 − 3.98i)7-s + (2.80 + 0.360i)8-s + (0.0334 − 0.785i)10-s + (1.09 + 0.634i)11-s + (−2.69 − 2.39i)13-s + (−3.01 − 5.77i)14-s + (3.94 + 0.677i)16-s + (−1.16 − 2.01i)17-s + (4.57 − 2.63i)19-s + (0.0945 − 1.10i)20-s + (1.51 + 0.963i)22-s + (0.778 − 1.34i)23-s + ⋯
L(s)  = 1  + (0.999 + 0.0425i)2-s + (0.996 + 0.0850i)4-s − 0.248i·5-s + (−0.870 − 1.50i)7-s + (0.991 + 0.127i)8-s + (0.0105 − 0.248i)10-s + (0.331 + 0.191i)11-s + (−0.748 − 0.662i)13-s + (−0.805 − 1.54i)14-s + (0.985 + 0.169i)16-s + (−0.281 − 0.488i)17-s + (1.04 − 0.605i)19-s + (0.0211 − 0.247i)20-s + (0.323 + 0.205i)22-s + (0.162 − 0.281i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.435 + 0.900i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.435 + 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
Sign: $0.435 + 0.900i$
Analytic conductor: \(7.47399\)
Root analytic conductor: \(2.73386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{936} (685, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 936,\ (\ :1/2),\ 0.435 + 0.900i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.29456 - 1.43967i\)
\(L(\frac12)\) \(\approx\) \(2.29456 - 1.43967i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.41 - 0.0601i)T \)
3 \( 1 \)
13 \( 1 + (2.69 + 2.39i)T \)
good5 \( 1 + 0.556iT - 5T^{2} \)
7 \( 1 + (2.30 + 3.98i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.09 - 0.634i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.16 + 2.01i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.57 + 2.63i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.778 + 1.34i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.923 - 0.533i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 7.00T + 31T^{2} \)
37 \( 1 + (-2.43 - 1.40i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.44 - 5.97i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.72 + 3.30i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 6.29T + 47T^{2} \)
53 \( 1 - 11.3iT - 53T^{2} \)
59 \( 1 + (-12.1 + 6.99i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.41 + 1.97i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.16 - 1.25i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (0.152 + 0.264i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 7.67T + 73T^{2} \)
79 \( 1 - 8.75T + 79T^{2} \)
83 \( 1 - 6.78iT - 83T^{2} \)
89 \( 1 + (6.28 - 10.8i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.36 - 2.36i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.06231768680575216106495646351, −9.295579723928322858958982014722, −7.84156796087404046098845547871, −7.10095208122050178432290314029, −6.64234599716886507451060405053, −5.34223510498457817076833688884, −4.58215001946788592911164025824, −3.60892718266820873288385646225, −2.75140521676788016949786858225, −0.940515043356385886496145483680, 1.95783702520390786548217330994, 2.92265332826005120320157845767, 3.78876963253004572573936777417, 5.11126449834477211236553517307, 5.78355710049625828151004017682, 6.60166475990552575183377827258, 7.34681938669274575744521639385, 8.623243298685826944619446216672, 9.448367481656890932459784175153, 10.22679652545511961551080303938

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