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Properties

Label 2-399-133.10-c1-0-21
Degree $2$
Conductor $399$
Sign $0.750 + 0.661i$
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  + (0.416 + 0.496i)2-s + (−0.173 − 0.984i)3-s + (0.274 − 1.55i)4-s + (2.05 − 0.362i)5-s + (0.416 − 0.496i)6-s + (2.34 + 1.22i)7-s + (2.00 − 1.15i)8-s + (−0.939 + 0.342i)9-s + (1.03 + 0.868i)10-s − 1.90·11-s − 1.58·12-s + (−2.61 − 2.19i)13-s + (0.370 + 1.67i)14-s + (−0.713 − 1.96i)15-s + (−1.55 − 0.566i)16-s + (−1.24 + 3.40i)17-s + ⋯
L(s)  = 1  + (0.294 + 0.351i)2-s + (−0.100 − 0.568i)3-s + (0.137 − 0.778i)4-s + (0.918 − 0.162i)5-s + (0.170 − 0.202i)6-s + (0.886 + 0.462i)7-s + (0.710 − 0.410i)8-s + (−0.313 + 0.114i)9-s + (0.327 + 0.274i)10-s − 0.574·11-s − 0.456·12-s + (−0.726 − 0.609i)13-s + (0.0989 + 0.447i)14-s + (−0.184 − 0.506i)15-s + (−0.389 − 0.141i)16-s + (−0.300 + 0.826i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 399 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.750 + 0.661i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 399 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.750 + 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(399\)    =    \(3 \cdot 7 \cdot 19\)
Sign: $0.750 + 0.661i$
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: \((2,\ 399,\ (\ :1/2),\ 0.750 + 0.661i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.79096 - 0.676867i\)
\(L(\frac12)\) \(\approx\) \(1.79096 - 0.676867i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.34 - 1.22i)T \)
19 \( 1 + (-4.32 + 0.571i)T \)
good2 \( 1 + (-0.416 - 0.496i)T + (-0.347 + 1.96i)T^{2} \)
5 \( 1 + (-2.05 + 0.362i)T + (4.69 - 1.71i)T^{2} \)
11 \( 1 + 1.90T + 11T^{2} \)
13 \( 1 + (2.61 + 2.19i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (1.24 - 3.40i)T + (-13.0 - 10.9i)T^{2} \)
23 \( 1 + (-0.609 - 0.511i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (-6.37 - 1.12i)T + (27.2 + 9.91i)T^{2} \)
31 \( 1 + (1.92 + 3.33i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-8.52 + 4.91i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (7.97 - 6.69i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (10.9 + 3.98i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (-3.43 - 9.42i)T + (-36.0 + 30.2i)T^{2} \)
53 \( 1 + (3.72 + 0.656i)T + (49.8 + 18.1i)T^{2} \)
59 \( 1 + (-7.14 - 2.60i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (5.19 - 6.19i)T + (-10.5 - 60.0i)T^{2} \)
67 \( 1 + (2.31 - 2.75i)T + (-11.6 - 65.9i)T^{2} \)
71 \( 1 + (-0.643 + 1.76i)T + (-54.3 - 45.6i)T^{2} \)
73 \( 1 + (11.7 - 2.07i)T + (68.5 - 24.9i)T^{2} \)
79 \( 1 + (-1.35 + 3.73i)T + (-60.5 - 50.7i)T^{2} \)
83 \( 1 + (-12.8 - 7.41i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (2.21 - 12.5i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (-0.898 - 5.09i)T + (-91.1 + 33.1i)T^{2} \)
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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

−11.14495669550893953147100492192, −10.25756735782925192637839219907, −9.471970259077477863306357017739, −8.228985837206708164201614234249, −7.35471892874652181965958992812, −6.16593847285590179163490702347, −5.48817699994348364797980409170, −4.77052190271132703781979835498, −2.51441044585199342054729154857, −1.40052863878226326105869461778, 2.03737072079354878606806379276, 3.17853025536294743621675892013, 4.60606871543162862499120733003, 5.16495080008316160685643056424, 6.73741665569304779458643423637, 7.67623441302718294127565200014, 8.665499758619108475232479826588, 9.822556377608461778443702630725, 10.45961122162349626465723756266, 11.55660840295033132509021428691

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