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Properties

Label 2-416-416.395-c1-0-41
Degree $2$
Conductor $416$
Sign $-0.951 + 0.307i$
Analytic cond. $3.32177$
Root an. cond. $1.82257$
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  + (−1.41 + 0.0124i)2-s + (0.185 + 0.447i)3-s + (1.99 − 0.0352i)4-s + (−0.172 − 0.416i)5-s + (−0.267 − 0.630i)6-s − 2.63·7-s + (−2.82 + 0.0748i)8-s + (1.95 − 1.95i)9-s + (0.249 + 0.586i)10-s + (−3.87 + 1.60i)11-s + (0.386 + 0.888i)12-s + (−3.51 − 0.795i)13-s + (3.72 − 0.0328i)14-s + (0.154 − 0.154i)15-s + (3.99 − 0.141i)16-s − 3.57·17-s + ⋯
L(s)  = 1  + (−0.999 + 0.00882i)2-s + (0.107 + 0.258i)3-s + (0.999 − 0.0176i)4-s + (−0.0771 − 0.186i)5-s + (−0.109 − 0.257i)6-s − 0.995·7-s + (−0.999 + 0.0264i)8-s + (0.651 − 0.651i)9-s + (0.0787 + 0.185i)10-s + (−1.16 + 0.484i)11-s + (0.111 + 0.256i)12-s + (−0.975 − 0.220i)13-s + (0.995 − 0.00878i)14-s + (0.0398 − 0.0398i)15-s + (0.999 − 0.0352i)16-s − 0.867·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.951 + 0.307i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.951 + 0.307i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(416\)    =    \(2^{5} \cdot 13\)
Sign: $-0.951 + 0.307i$
Analytic conductor: \(3.32177\)
Root analytic conductor: \(1.82257\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{416} (395, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 416,\ (\ :1/2),\ -0.951 + 0.307i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0145952 - 0.0927568i\)
\(L(\frac12)\) \(\approx\) \(0.0145952 - 0.0927568i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.41 - 0.0124i)T \)
13 \( 1 + (3.51 + 0.795i)T \)
good3 \( 1 + (-0.185 - 0.447i)T + (-2.12 + 2.12i)T^{2} \)
5 \( 1 + (0.172 + 0.416i)T + (-3.53 + 3.53i)T^{2} \)
7 \( 1 + 2.63T + 7T^{2} \)
11 \( 1 + (3.87 - 1.60i)T + (7.77 - 7.77i)T^{2} \)
17 \( 1 + 3.57T + 17T^{2} \)
19 \( 1 + (0.249 - 0.603i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (2.69 - 2.69i)T - 23iT^{2} \)
29 \( 1 + (1.95 - 0.808i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 + (3.43 + 3.43i)T + 31iT^{2} \)
37 \( 1 + (2.19 - 0.907i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 - 0.550iT - 41T^{2} \)
43 \( 1 + (4.93 + 2.04i)T + (30.4 + 30.4i)T^{2} \)
47 \( 1 + (2.63 + 2.63i)T + 47iT^{2} \)
53 \( 1 + (11.0 + 4.58i)T + (37.4 + 37.4i)T^{2} \)
59 \( 1 + (0.722 + 1.74i)T + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (-2.34 + 0.973i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 + (-10.2 - 4.23i)T + (47.3 + 47.3i)T^{2} \)
71 \( 1 - 0.852iT - 71T^{2} \)
73 \( 1 - 6.22T + 73T^{2} \)
79 \( 1 + 1.67T + 79T^{2} \)
83 \( 1 + (3.20 - 7.73i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 - 3.24iT - 89T^{2} \)
97 \( 1 + (-13.6 + 13.6i)T - 97iT^{2} \)
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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

−10.37378084258214575161844734619, −9.879504411165509557723557535463, −9.218664100042432537500691007969, −8.096685403730316685307567256924, −7.16689124799014682962208946087, −6.39876696096767270380128798445, −5.01814173886015846097775511263, −3.48714348112842098169378105048, −2.23588601723039665499962157368, −0.07380022883093821362308396263, 2.12015535887364668814730390502, 3.15222514264820210374902602688, 4.97078178421053723326455733696, 6.35207339830799185622300587889, 7.15661789449653373788506057003, 7.893695008413628642795489632245, 8.930631109901544450764773888446, 9.868196627446215161660033462273, 10.53453925001558907679997214274, 11.29834095592125375872089879701

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