Properties

Label 2-392-392.37-c1-0-15
Degree $2$
Conductor $392$
Sign $0.0118 + 0.999i$
Analytic cond. $3.13013$
Root an. cond. $1.76921$
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  + (−0.767 − 1.18i)2-s + (−1.83 − 1.98i)3-s + (−0.821 + 1.82i)4-s + (−0.819 + 2.65i)5-s + (−0.943 + 3.70i)6-s + (2.51 − 0.807i)7-s + (2.79 − 0.423i)8-s + (−0.322 + 4.30i)9-s + (3.78 − 1.06i)10-s + (0.808 − 0.0606i)11-s + (5.12 − 1.72i)12-s + (0.864 − 1.79i)13-s + (−2.89 − 2.37i)14-s + (6.77 − 3.26i)15-s + (−2.64 − 2.99i)16-s + (−2.72 − 0.410i)17-s + ⋯
L(s)  = 1  + (−0.542 − 0.839i)2-s + (−1.06 − 1.14i)3-s + (−0.410 + 0.911i)4-s + (−0.366 + 1.18i)5-s + (−0.384 + 1.51i)6-s + (0.952 − 0.305i)7-s + (0.988 − 0.149i)8-s + (−0.107 + 1.43i)9-s + (1.19 − 0.337i)10-s + (0.243 − 0.0182i)11-s + (1.48 − 0.498i)12-s + (0.239 − 0.497i)13-s + (−0.773 − 0.634i)14-s + (1.75 − 0.842i)15-s + (−0.662 − 0.749i)16-s + (−0.661 − 0.0996i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0118 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0118 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(392\)    =    \(2^{3} \cdot 7^{2}\)
Sign: $0.0118 + 0.999i$
Analytic conductor: \(3.13013\)
Root analytic conductor: \(1.76921\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{392} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 392,\ (\ :1/2),\ 0.0118 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.516404 - 0.510295i\)
\(L(\frac12)\) \(\approx\) \(0.516404 - 0.510295i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.767 + 1.18i)T \)
7 \( 1 + (-2.51 + 0.807i)T \)
good3 \( 1 + (1.83 + 1.98i)T + (-0.224 + 2.99i)T^{2} \)
5 \( 1 + (0.819 - 2.65i)T + (-4.13 - 2.81i)T^{2} \)
11 \( 1 + (-0.808 + 0.0606i)T + (10.8 - 1.63i)T^{2} \)
13 \( 1 + (-0.864 + 1.79i)T + (-8.10 - 10.1i)T^{2} \)
17 \( 1 + (2.72 + 0.410i)T + (16.2 + 5.01i)T^{2} \)
19 \( 1 + (-5.29 - 3.05i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-5.21 + 0.786i)T + (21.9 - 6.77i)T^{2} \)
29 \( 1 + (1.35 + 1.08i)T + (6.45 + 28.2i)T^{2} \)
31 \( 1 + (4.70 + 8.14i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-10.2 + 4.02i)T + (27.1 - 25.1i)T^{2} \)
41 \( 1 + (-0.479 - 2.09i)T + (-36.9 + 17.7i)T^{2} \)
43 \( 1 + (1.35 + 0.309i)T + (38.7 + 18.6i)T^{2} \)
47 \( 1 + (-4.15 + 2.83i)T + (17.1 - 43.7i)T^{2} \)
53 \( 1 + (-2.81 - 1.10i)T + (38.8 + 36.0i)T^{2} \)
59 \( 1 + (2.78 + 9.02i)T + (-48.7 + 33.2i)T^{2} \)
61 \( 1 + (6.37 - 2.50i)T + (44.7 - 41.4i)T^{2} \)
67 \( 1 + (-13.1 + 7.59i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-8.16 - 10.2i)T + (-15.7 + 69.2i)T^{2} \)
73 \( 1 + (-9.40 - 6.41i)T + (26.6 + 67.9i)T^{2} \)
79 \( 1 + (-5.74 + 9.95i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.46 - 5.11i)T + (-51.7 + 64.8i)T^{2} \)
89 \( 1 + (0.395 - 5.27i)T + (-88.0 - 13.2i)T^{2} \)
97 \( 1 - 5.90T + 97T^{2} \)
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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

−11.22633944954206769368523286923, −10.71924634196442898299779175368, −9.445421196210376111250763341153, −7.914529695132782432801214120944, −7.51173407363852446324966300928, −6.60975221997423870115905734924, −5.30636080061722273364670790102, −3.79853811253057311039153814080, −2.32159323707970267794763044920, −0.910645572578480611034143711054, 1.08537713108171006682374477906, 4.22659942249210842729573773879, 4.97795803751068358908797250862, 5.40995561302858160869384755275, 6.73301311867198180044917865741, 7.965390331050163152590296094180, 9.141742306340744451266574361682, 9.231845732757796168962799076820, 10.71990249467062195919201138594, 11.25649501567174945006711053261

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