Properties

Label 2-392-392.37-c1-0-21
Degree $2$
Conductor $392$
Sign $0.950 + 0.309i$
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  + (−1.19 − 0.751i)2-s + (0.581 + 0.626i)3-s + (0.871 + 1.80i)4-s + (−0.147 + 0.478i)5-s + (−0.225 − 1.18i)6-s + (2.54 − 0.722i)7-s + (0.307 − 2.81i)8-s + (0.169 − 2.26i)9-s + (0.536 − 0.462i)10-s + (−1.85 + 0.138i)11-s + (−0.620 + 1.59i)12-s + (0.132 − 0.275i)13-s + (−3.59 − 1.04i)14-s + (−0.385 + 0.185i)15-s + (−2.48 + 3.13i)16-s + (5.30 + 0.800i)17-s + ⋯
L(s)  = 1  + (−0.847 − 0.531i)2-s + (0.335 + 0.361i)3-s + (0.435 + 0.900i)4-s + (−0.0660 + 0.214i)5-s + (−0.0922 − 0.484i)6-s + (0.962 − 0.272i)7-s + (0.108 − 0.994i)8-s + (0.0565 − 0.754i)9-s + (0.169 − 0.146i)10-s + (−0.557 + 0.0418i)11-s + (−0.179 + 0.459i)12-s + (0.0367 − 0.0763i)13-s + (−0.960 − 0.279i)14-s + (−0.0995 + 0.0479i)15-s + (−0.620 + 0.784i)16-s + (1.28 + 0.194i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.950 + 0.309i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.950 + 0.309i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(392\)    =    \(2^{3} \cdot 7^{2}\)
Sign: $0.950 + 0.309i$
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: \((2,\ 392,\ (\ :1/2),\ 0.950 + 0.309i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.13231 - 0.179688i\)
\(L(\frac12)\) \(\approx\) \(1.13231 - 0.179688i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.19 + 0.751i)T \)
7 \( 1 + (-2.54 + 0.722i)T \)
good3 \( 1 + (-0.581 - 0.626i)T + (-0.224 + 2.99i)T^{2} \)
5 \( 1 + (0.147 - 0.478i)T + (-4.13 - 2.81i)T^{2} \)
11 \( 1 + (1.85 - 0.138i)T + (10.8 - 1.63i)T^{2} \)
13 \( 1 + (-0.132 + 0.275i)T + (-8.10 - 10.1i)T^{2} \)
17 \( 1 + (-5.30 - 0.800i)T + (16.2 + 5.01i)T^{2} \)
19 \( 1 + (-2.82 - 1.63i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.54 + 0.232i)T + (21.9 - 6.77i)T^{2} \)
29 \( 1 + (1.53 + 1.22i)T + (6.45 + 28.2i)T^{2} \)
31 \( 1 + (-1.25 - 2.16i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (7.96 - 3.12i)T + (27.1 - 25.1i)T^{2} \)
41 \( 1 + (2.42 + 10.6i)T + (-36.9 + 17.7i)T^{2} \)
43 \( 1 + (-11.6 - 2.66i)T + (38.7 + 18.6i)T^{2} \)
47 \( 1 + (5.80 - 3.95i)T + (17.1 - 43.7i)T^{2} \)
53 \( 1 + (8.58 + 3.36i)T + (38.8 + 36.0i)T^{2} \)
59 \( 1 + (-2.05 - 6.65i)T + (-48.7 + 33.2i)T^{2} \)
61 \( 1 + (-8.93 + 3.50i)T + (44.7 - 41.4i)T^{2} \)
67 \( 1 + (-1.84 + 1.06i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3.67 - 4.61i)T + (-15.7 + 69.2i)T^{2} \)
73 \( 1 + (9.40 + 6.41i)T + (26.6 + 67.9i)T^{2} \)
79 \( 1 + (3.32 - 5.75i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.776 - 1.61i)T + (-51.7 + 64.8i)T^{2} \)
89 \( 1 + (0.590 - 7.87i)T + (-88.0 - 13.2i)T^{2} \)
97 \( 1 + 4.65T + 97T^{2} \)
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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

−11.03323631081446146504187046546, −10.32677024094367127019177791908, −9.528688300882392076319360760312, −8.564098813219727442824717422640, −7.78915220925899380016828857294, −6.92655585995993724405800878163, −5.34493299553483338283674753836, −3.89852685534825009061298944448, −2.95832644926530690507798898724, −1.27394302183474900395103304228, 1.35554782634605526423099239804, 2.68064449736856827803764133623, 4.89666935270185337123028475740, 5.52529056530402487971933204787, 7.01807547499102584563645400398, 7.84263234548181351400011227275, 8.325802783972528831660825195243, 9.328309538268455543806671196266, 10.39126434066245607887661578453, 11.13442462279387430953788113829

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