Properties

Label 2-392-392.299-c1-0-21
Degree $2$
Conductor $392$
Sign $0.660 - 0.751i$
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  + (−1.04 + 0.949i)2-s + (1.12 − 1.65i)3-s + (0.198 − 1.99i)4-s + (−0.117 + 1.56i)5-s + (0.388 + 2.80i)6-s + (1.24 + 2.33i)7-s + (1.68 + 2.27i)8-s + (−0.373 − 0.952i)9-s + (−1.36 − 1.74i)10-s + (−0.587 + 1.49i)11-s + (−3.07 − 2.57i)12-s + (2.03 + 2.55i)13-s + (−3.52 − 1.26i)14-s + (2.45 + 1.95i)15-s + (−3.92 − 0.789i)16-s + (−4.02 − 4.33i)17-s + ⋯
L(s)  = 1  + (−0.741 + 0.671i)2-s + (0.652 − 0.956i)3-s + (0.0991 − 0.995i)4-s + (−0.0523 + 0.698i)5-s + (0.158 + 1.14i)6-s + (0.471 + 0.881i)7-s + (0.594 + 0.804i)8-s + (−0.124 − 0.317i)9-s + (−0.430 − 0.553i)10-s + (−0.176 + 0.450i)11-s + (−0.887 − 0.743i)12-s + (0.564 + 0.707i)13-s + (−0.941 − 0.336i)14-s + (0.634 + 0.505i)15-s + (−0.980 − 0.197i)16-s + (−0.976 − 1.05i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.660 - 0.751i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.660 - 0.751i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(392\)    =    \(2^{3} \cdot 7^{2}\)
Sign: $0.660 - 0.751i$
Analytic conductor: \(3.13013\)
Root analytic conductor: \(1.76921\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{392} (299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 392,\ (\ :1/2),\ 0.660 - 0.751i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.11294 + 0.503636i\)
\(L(\frac12)\) \(\approx\) \(1.11294 + 0.503636i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.04 - 0.949i)T \)
7 \( 1 + (-1.24 - 2.33i)T \)
good3 \( 1 + (-1.12 + 1.65i)T + (-1.09 - 2.79i)T^{2} \)
5 \( 1 + (0.117 - 1.56i)T + (-4.94 - 0.745i)T^{2} \)
11 \( 1 + (0.587 - 1.49i)T + (-8.06 - 7.48i)T^{2} \)
13 \( 1 + (-2.03 - 2.55i)T + (-2.89 + 12.6i)T^{2} \)
17 \( 1 + (4.02 + 4.33i)T + (-1.27 + 16.9i)T^{2} \)
19 \( 1 + (-3.91 - 2.26i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.12 - 1.21i)T + (-1.71 - 22.9i)T^{2} \)
29 \( 1 + (-2.66 + 0.607i)T + (26.1 - 12.5i)T^{2} \)
31 \( 1 + (-1.55 - 2.69i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.01 + 6.51i)T + (-30.5 - 20.8i)T^{2} \)
41 \( 1 + (3.96 + 8.22i)T + (-25.5 + 32.0i)T^{2} \)
43 \( 1 + (-0.651 - 0.313i)T + (26.8 + 33.6i)T^{2} \)
47 \( 1 + (1.86 - 0.280i)T + (44.9 - 13.8i)T^{2} \)
53 \( 1 + (-2.26 - 7.32i)T + (-43.7 + 29.8i)T^{2} \)
59 \( 1 + (2.41 - 0.181i)T + (58.3 - 8.79i)T^{2} \)
61 \( 1 + (2.70 + 0.833i)T + (50.4 + 34.3i)T^{2} \)
67 \( 1 + (-3.54 - 6.13i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (11.5 + 2.63i)T + (63.9 + 30.8i)T^{2} \)
73 \( 1 + (-1.81 + 12.0i)T + (-69.7 - 21.5i)T^{2} \)
79 \( 1 + (13.2 + 7.63i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.108 - 0.0868i)T + (18.4 + 80.9i)T^{2} \)
89 \( 1 + (-14.7 + 5.79i)T + (65.2 - 60.5i)T^{2} \)
97 \( 1 + 6.61iT - 97T^{2} \)
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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

−11.36557823176393329244779524201, −10.41729622811466263863945344875, −9.180959928667063130344165905338, −8.644080304744168342418013481100, −7.55574150781716661774615618850, −7.07235398968897357632361200285, −6.06969200275870926205068108842, −4.82205116731363837821129301216, −2.69577959493706666025494510189, −1.69530794436976896002472064557, 1.09586870157353784650386727905, 2.98626121525500567248123970366, 3.99812684514931813919360425466, 4.81266304900630471433067695429, 6.66323985700676869049810909995, 8.176338252885787426605262389443, 8.398769416219257086132876981669, 9.423582836401271008061853814811, 10.26220811337871584770777411350, 10.87365572146106539919842891633

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