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Properties

Label 2-392-392.109-c1-0-28
Degree $2$
Conductor $392$
Sign $-0.0347 - 0.999i$
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  + (1.02 + 0.976i)2-s + (0.131 + 0.869i)3-s + (0.0930 + 1.99i)4-s + (0.989 + 0.388i)5-s + (−0.715 + 1.01i)6-s + (2.48 − 0.916i)7-s + (−1.85 + 2.13i)8-s + (2.12 − 0.656i)9-s + (0.633 + 1.36i)10-s + (0.522 − 1.69i)11-s + (−1.72 + 0.342i)12-s + (−0.782 + 0.178i)13-s + (3.43 + 1.48i)14-s + (−0.208 + 0.911i)15-s + (−3.98 + 0.371i)16-s + (−0.935 + 0.637i)17-s + ⋯
L(s)  = 1  + (0.723 + 0.690i)2-s + (0.0756 + 0.502i)3-s + (0.0465 + 0.998i)4-s + (0.442 + 0.173i)5-s + (−0.292 + 0.415i)6-s + (0.938 − 0.346i)7-s + (−0.656 + 0.754i)8-s + (0.709 − 0.218i)9-s + (0.200 + 0.431i)10-s + (0.157 − 0.510i)11-s + (−0.498 + 0.0989i)12-s + (−0.217 + 0.0495i)13-s + (0.917 + 0.397i)14-s + (−0.0537 + 0.235i)15-s + (−0.995 + 0.0929i)16-s + (−0.226 + 0.154i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0347 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0347 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(392\)    =    \(2^{3} \cdot 7^{2}\)
Sign: $-0.0347 - 0.999i$
Analytic conductor: \(3.13013\)
Root analytic conductor: \(1.76921\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{392} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 392,\ (\ :1/2),\ -0.0347 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.60143 + 1.65811i\)
\(L(\frac12)\) \(\approx\) \(1.60143 + 1.65811i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.02 - 0.976i)T \)
7 \( 1 + (-2.48 + 0.916i)T \)
good3 \( 1 + (-0.131 - 0.869i)T + (-2.86 + 0.884i)T^{2} \)
5 \( 1 + (-0.989 - 0.388i)T + (3.66 + 3.40i)T^{2} \)
11 \( 1 + (-0.522 + 1.69i)T + (-9.08 - 6.19i)T^{2} \)
13 \( 1 + (0.782 - 0.178i)T + (11.7 - 5.64i)T^{2} \)
17 \( 1 + (0.935 - 0.637i)T + (6.21 - 15.8i)T^{2} \)
19 \( 1 + (6.50 - 3.75i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.67 + 3.18i)T + (8.40 + 21.4i)T^{2} \)
29 \( 1 + (0.367 - 0.762i)T + (-18.0 - 22.6i)T^{2} \)
31 \( 1 + (-4.28 + 7.42i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-8.15 + 0.611i)T + (36.5 - 5.51i)T^{2} \)
41 \( 1 + (-3.46 - 4.34i)T + (-9.12 + 39.9i)T^{2} \)
43 \( 1 + (7.41 + 5.91i)T + (9.56 + 41.9i)T^{2} \)
47 \( 1 + (2.23 - 2.06i)T + (3.51 - 46.8i)T^{2} \)
53 \( 1 + (-10.0 - 0.755i)T + (52.4 + 7.89i)T^{2} \)
59 \( 1 + (6.39 - 2.51i)T + (43.2 - 40.1i)T^{2} \)
61 \( 1 + (1.48 - 0.111i)T + (60.3 - 9.09i)T^{2} \)
67 \( 1 + (-0.789 - 0.455i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-8.57 + 4.12i)T + (44.2 - 55.5i)T^{2} \)
73 \( 1 + (0.789 + 0.732i)T + (5.45 + 72.7i)T^{2} \)
79 \( 1 + (-5.34 - 9.25i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (12.6 + 2.88i)T + (74.7 + 36.0i)T^{2} \)
89 \( 1 + (15.8 - 4.87i)T + (73.5 - 50.1i)T^{2} \)
97 \( 1 - 11.7T + 97T^{2} \)
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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

−11.62440116307342899319477694448, −10.64003003531388453709591986905, −9.769084384231101690266877110394, −8.486018371994280825252860599970, −7.84878890551655074649810285012, −6.58935687958026563145835912319, −5.83685073945078532697730925150, −4.44918145757825168879520039089, −4.01256250031839351804219752415, −2.22193979615730806604904309299, 1.57245361146619566162603924911, 2.38656768563116301519637797861, 4.23492654778914357328588647678, 4.97181345891336415203822127611, 6.14714840891428900317223352212, 7.14782730114205694128043324248, 8.349347921708129793950629597927, 9.481502861069321017338608484629, 10.29896821329540308779422595386, 11.26843788139825748890973947322

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