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Properties

Label 2-399-133.10-c1-0-5
Degree $2$
Conductor $399$
Sign $0.0103 + 0.999i$
Analytic cond. $3.18603$
Root an. cond. $1.78494$
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  + (−1.74 − 2.08i)2-s + (−0.173 − 0.984i)3-s + (−0.935 + 5.30i)4-s + (−3.54 + 0.625i)5-s + (−1.74 + 2.08i)6-s + (−0.498 + 2.59i)7-s + (7.97 − 4.60i)8-s + (−0.939 + 0.342i)9-s + (7.50 + 6.29i)10-s + 1.37·11-s + 5.38·12-s + (−2.65 − 2.23i)13-s + (6.28 − 3.50i)14-s + (1.23 + 3.38i)15-s + (−13.3 − 4.87i)16-s + (1.18 − 3.24i)17-s + ⋯
L(s)  = 1  + (−1.23 − 1.47i)2-s + (−0.100 − 0.568i)3-s + (−0.467 + 2.65i)4-s + (−1.58 + 0.279i)5-s + (−0.713 + 0.850i)6-s + (−0.188 + 0.982i)7-s + (2.81 − 1.62i)8-s + (−0.313 + 0.114i)9-s + (2.37 + 1.99i)10-s + 0.415·11-s + 1.55·12-s + (−0.737 − 0.618i)13-s + (1.67 − 0.935i)14-s + (0.318 + 0.874i)15-s + (−3.34 − 1.21i)16-s + (0.286 − 0.787i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 399 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0103 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 399 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0103 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(399\)    =    \(3 \cdot 7 \cdot 19\)
Sign: $0.0103 + 0.999i$
Analytic conductor: \(3.18603\)
Root analytic conductor: \(1.78494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{399} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 399,\ (\ :1/2),\ 0.0103 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.308923 - 0.305737i\)
\(L(\frac12)\) \(\approx\) \(0.308923 - 0.305737i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.173 + 0.984i)T \)
7 \( 1 + (0.498 - 2.59i)T \)
19 \( 1 + (-3.66 + 2.36i)T \)
good2 \( 1 + (1.74 + 2.08i)T + (-0.347 + 1.96i)T^{2} \)
5 \( 1 + (3.54 - 0.625i)T + (4.69 - 1.71i)T^{2} \)
11 \( 1 - 1.37T + 11T^{2} \)
13 \( 1 + (2.65 + 2.23i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (-1.18 + 3.24i)T + (-13.0 - 10.9i)T^{2} \)
23 \( 1 + (-0.256 - 0.214i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (-3.92 - 0.692i)T + (27.2 + 9.91i)T^{2} \)
31 \( 1 + (-5.13 - 8.88i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-6.84 + 3.95i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (5.23 - 4.39i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (-5.95 - 2.16i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (1.03 + 2.85i)T + (-36.0 + 30.2i)T^{2} \)
53 \( 1 + (-6.37 - 1.12i)T + (49.8 + 18.1i)T^{2} \)
59 \( 1 + (1.10 + 0.402i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-2.61 + 3.11i)T + (-10.5 - 60.0i)T^{2} \)
67 \( 1 + (-4.37 + 5.20i)T + (-11.6 - 65.9i)T^{2} \)
71 \( 1 + (2.67 - 7.35i)T + (-54.3 - 45.6i)T^{2} \)
73 \( 1 + (1.59 - 0.280i)T + (68.5 - 24.9i)T^{2} \)
79 \( 1 + (0.0888 - 0.244i)T + (-60.5 - 50.7i)T^{2} \)
83 \( 1 + (-5.80 - 3.34i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-2.42 + 13.7i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (0.729 + 4.13i)T + (-91.1 + 33.1i)T^{2} \)
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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

−11.31170539839533529377994832725, −10.23157168311052165016288215088, −9.243878045362014966993149802335, −8.409082156953390472819723780156, −7.69582595828402712375605330649, −6.93426200720374123451036592063, −4.79623808051902378955804246355, −3.30073405043349277432364054408, −2.66833245802991167120059959470, −0.76123569881574235759993973224, 0.76109614301385904870566468338, 3.98031272873911119903422917526, 4.72422646115251865892209967898, 6.12197864388291769111916491673, 7.16615476427279130545045272683, 7.79243133145363022215444834538, 8.491313636679853440417369676711, 9.579320695005765300604798020383, 10.19256987611226080332526805239, 11.20037100615378240504429628292

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