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Properties

Label 2-399-7.2-c1-0-16
Degree $2$
Conductor $399$
Sign $0.100 + 0.994i$
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  + (0.676 − 1.17i)2-s + (−0.5 − 0.866i)3-s + (0.0838 + 0.145i)4-s + (0.583 − 1.01i)5-s − 1.35·6-s + (1.40 + 2.24i)7-s + 2.93·8-s + (−0.499 + 0.866i)9-s + (−0.790 − 1.36i)10-s + (−3.00 − 5.20i)11-s + (0.0838 − 0.145i)12-s + 6.00·13-s + (3.57 − 0.132i)14-s − 1.16·15-s + (1.81 − 3.14i)16-s + (−1.5 − 2.59i)17-s + ⋯
L(s)  = 1  + (0.478 − 0.828i)2-s + (−0.288 − 0.499i)3-s + (0.0419 + 0.0725i)4-s + (0.261 − 0.452i)5-s − 0.552·6-s + (0.531 + 0.846i)7-s + 1.03·8-s + (−0.166 + 0.288i)9-s + (−0.249 − 0.432i)10-s + (−0.905 − 1.56i)11-s + (0.0241 − 0.0419i)12-s + 1.66·13-s + (0.956 − 0.0355i)14-s − 0.301·15-s + (0.454 − 0.787i)16-s + (−0.363 − 0.630i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 399 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.100 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 399 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.100 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(399\)    =    \(3 \cdot 7 \cdot 19\)
Sign: $0.100 + 0.994i$
Analytic conductor: \(3.18603\)
Root analytic conductor: \(1.78494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{399} (58, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 399,\ (\ :1/2),\ 0.100 + 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.41416 - 1.27875i\)
\(L(\frac12)\) \(\approx\) \(1.41416 - 1.27875i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (-1.40 - 2.24i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (-0.676 + 1.17i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (-0.583 + 1.01i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (3.00 + 5.20i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 6.00T + 13T^{2} \)
17 \( 1 + (1.5 + 2.59i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (3.24 - 5.61i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 6.13T + 29T^{2} \)
31 \( 1 + (3.73 + 6.47i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.82 - 3.15i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 4.77T + 41T^{2} \)
43 \( 1 + 0.507T + 43T^{2} \)
47 \( 1 + (-2.68 + 4.64i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.65 - 6.32i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.44 - 7.69i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.670 - 1.16i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.18 - 3.78i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 9.92T + 71T^{2} \)
73 \( 1 + (1.94 + 3.36i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.34 - 9.25i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 3.07T + 83T^{2} \)
89 \( 1 + (4.52 - 7.84i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 1.01T + 97T^{2} \)
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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

−11.35952779457341562436656901196, −10.70216759495511674388134809487, −9.129085213959679608489678914731, −8.343549499600599147441256650883, −7.47288209199241874365150535971, −5.83195245891497576381909975016, −5.40746633629249302237102942028, −3.83024396406843420419434859585, −2.66646155700072771625035334541, −1.37845946394378411367482400541, 1.84755085270919642383975048142, 3.93731559895141115478543231516, 4.71756908049138851120169371514, 5.77966728844986376465364917432, 6.67419952732410032522352563343, 7.48949172030391331250162572302, 8.514649475607647953949101341148, 10.04836624544270733273147257636, 10.59563955945154794593670533975, 11.07650188316144119457050417513

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