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Properties

Label 2-392-7.4-c3-0-7
Degree $2$
Conductor $392$
Sign $-0.605 - 0.795i$
Analytic cond. $23.1287$
Root an. cond. $4.80923$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.27 + 5.67i)3-s + (−1.72 − 2.98i)5-s + (−7.95 − 13.7i)9-s + (13.6 − 23.6i)11-s + 58.7·13-s + 22.5·15-s + (−24.6 + 42.6i)17-s + (78.5 + 136. i)19-s + (41.0 + 71.1i)23-s + (56.5 − 97.9i)25-s − 72.7·27-s − 194.·29-s + (−57.8 + 100. i)31-s + (89.4 + 154. i)33-s + (−163. − 283. i)37-s + ⋯
L(s)  = 1  + (−0.630 + 1.09i)3-s + (−0.154 − 0.267i)5-s + (−0.294 − 0.510i)9-s + (0.374 − 0.648i)11-s + 1.25·13-s + 0.388·15-s + (−0.351 + 0.609i)17-s + (0.948 + 1.64i)19-s + (0.372 + 0.645i)23-s + (0.452 − 0.783i)25-s − 0.518·27-s − 1.24·29-s + (−0.335 + 0.580i)31-s + (0.471 + 0.816i)33-s + (−0.727 − 1.26i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(392\)    =    \(2^{3} \cdot 7^{2}\)
Sign: $-0.605 - 0.795i$
Analytic conductor: \(23.1287\)
Root analytic conductor: \(4.80923\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{392} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 392,\ (\ :3/2),\ -0.605 - 0.795i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.243516511\)
\(L(\frac12)\) \(\approx\) \(1.243516511\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (3.27 - 5.67i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 + (1.72 + 2.98i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-13.6 + 23.6i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 58.7T + 2.19e3T^{2} \)
17 \( 1 + (24.6 - 42.6i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-78.5 - 136. i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-41.0 - 71.1i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 194.T + 2.43e4T^{2} \)
31 \( 1 + (57.8 - 100. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (163. + 283. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 136.T + 6.89e4T^{2} \)
43 \( 1 - 311.T + 7.95e4T^{2} \)
47 \( 1 + (-177. - 308. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (338. - 586. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-98.5 + 170. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (30.5 + 52.8i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (508. - 881. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 279.T + 3.57e5T^{2} \)
73 \( 1 + (314. - 545. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (10.1 + 17.4i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 260.T + 5.71e5T^{2} \)
89 \( 1 + (-454. - 788. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 100.T + 9.12e5T^{2} \)
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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

−10.97114314323811628589599918619, −10.54164685559355826748142122610, −9.407213868712845976193899505853, −8.672754768667856250894058043085, −7.55287363335577231181683806240, −6.01762098693751720998503818622, −5.52414590073052368299060727894, −4.15252697017809086025306564062, −3.52646038245398167033135519179, −1.32488413754763664614822599262, 0.51044154890310628630668984637, 1.73936515948009783983402204108, 3.28889026516358943425843435245, 4.78801573495647768437565734280, 5.94437513584983046003321916670, 6.97543362889899955033211965105, 7.28694373467596359143036535519, 8.708001278951378759796370927721, 9.556705338441136949252620468584, 11.01490793572365672752793460126

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