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Properties

Label 2-392-392.251-c1-0-2
Degree $2$
Conductor $392$
Sign $-0.995 + 0.0978i$
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  + (1.39 − 0.247i)2-s + (−1.38 + 2.88i)3-s + (1.87 − 0.688i)4-s + (−3.27 − 1.57i)5-s + (−1.22 + 4.35i)6-s + (−2.59 + 0.490i)7-s + (2.44 − 1.42i)8-s + (−4.51 − 5.66i)9-s + (−4.95 − 1.38i)10-s + (−2.86 + 3.59i)11-s + (−0.623 + 6.37i)12-s + (−0.469 + 0.588i)13-s + (−3.49 + 1.32i)14-s + (9.09 − 7.25i)15-s + (3.05 − 2.58i)16-s + (−1.14 − 0.262i)17-s + ⋯
L(s)  = 1  + (0.984 − 0.174i)2-s + (−0.801 + 1.66i)3-s + (0.938 − 0.344i)4-s + (−1.46 − 0.705i)5-s + (−0.498 + 1.77i)6-s + (−0.982 + 0.185i)7-s + (0.864 − 0.502i)8-s + (−1.50 − 1.88i)9-s + (−1.56 − 0.438i)10-s + (−0.863 + 1.08i)11-s + (−0.180 + 1.83i)12-s + (−0.130 + 0.163i)13-s + (−0.935 + 0.354i)14-s + (2.34 − 1.87i)15-s + (0.763 − 0.646i)16-s + (−0.278 − 0.0635i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 + 0.0978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(392\)    =    \(2^{3} \cdot 7^{2}\)
Sign: $-0.995 + 0.0978i$
Analytic conductor: \(3.13013\)
Root analytic conductor: \(1.76921\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{392} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 392,\ (\ :1/2),\ -0.995 + 0.0978i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0225024 - 0.459006i\)
\(L(\frac12)\) \(\approx\) \(0.0225024 - 0.459006i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.39 + 0.247i)T \)
7 \( 1 + (2.59 - 0.490i)T \)
good3 \( 1 + (1.38 - 2.88i)T + (-1.87 - 2.34i)T^{2} \)
5 \( 1 + (3.27 + 1.57i)T + (3.11 + 3.90i)T^{2} \)
11 \( 1 + (2.86 - 3.59i)T + (-2.44 - 10.7i)T^{2} \)
13 \( 1 + (0.469 - 0.588i)T + (-2.89 - 12.6i)T^{2} \)
17 \( 1 + (1.14 + 0.262i)T + (15.3 + 7.37i)T^{2} \)
19 \( 1 - 3.38iT - 19T^{2} \)
23 \( 1 + (-0.198 + 0.0452i)T + (20.7 - 9.97i)T^{2} \)
29 \( 1 + (2.23 + 0.509i)T + (26.1 + 12.5i)T^{2} \)
31 \( 1 + 1.81T + 31T^{2} \)
37 \( 1 + (-4.22 - 0.965i)T + (33.3 + 16.0i)T^{2} \)
41 \( 1 + (3.57 - 7.42i)T + (-25.5 - 32.0i)T^{2} \)
43 \( 1 + (9.54 - 4.59i)T + (26.8 - 33.6i)T^{2} \)
47 \( 1 + (-2.37 + 2.97i)T + (-10.4 - 45.8i)T^{2} \)
53 \( 1 + (3.73 - 0.852i)T + (47.7 - 22.9i)T^{2} \)
59 \( 1 + (1.91 + 3.97i)T + (-36.7 + 46.1i)T^{2} \)
61 \( 1 + (2.62 - 11.4i)T + (-54.9 - 26.4i)T^{2} \)
67 \( 1 - 9.05T + 67T^{2} \)
71 \( 1 + (9.40 - 2.14i)T + (63.9 - 30.8i)T^{2} \)
73 \( 1 + (-9.57 + 7.63i)T + (16.2 - 71.1i)T^{2} \)
79 \( 1 - 10.7iT - 79T^{2} \)
83 \( 1 + (-6.26 + 4.99i)T + (18.4 - 80.9i)T^{2} \)
89 \( 1 + (2.96 - 2.36i)T + (19.8 - 86.7i)T^{2} \)
97 \( 1 + 0.471iT - 97T^{2} \)
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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

−11.82798141758853300012491438145, −11.03317200577539766067841182696, −10.11735830561101270203983683487, −9.470353502109625431779835388444, −8.048067415639882740874662193893, −6.76253698939149764057359268791, −5.54616437880815768594257866598, −4.71113920708745500847601834397, −4.08383815055407032459379422354, −3.16396947363534288683908028774, 0.22643599730258336863871902594, 2.61226657454704472139236536775, 3.55557547130546269098524348109, 5.21062606744264358529261811119, 6.25266826524366511832458902999, 6.94611471299617643886233063965, 7.56355661031255258368731317546, 8.341305046104354738351104407589, 10.73790670729385428676330880138, 11.11194582292941760448690986398

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