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Properties

Label 2-416-13.3-c1-0-1
Degree $2$
Conductor $416$
Sign $-0.923 - 0.384i$
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  + (1.17 + 2.04i)3-s − 2.56·5-s + (−1.84 + 3.18i)7-s + (−1.28 + 2.21i)9-s + (−0.516 − 0.895i)11-s + (−2.84 − 2.21i)13-s + (−3.02 − 5.23i)15-s + (−3.06 + 5.30i)17-s + (2.50 − 4.33i)19-s − 8.68·21-s + (3.53 + 6.12i)23-s + 1.56·25-s + 1.03·27-s + (2.5 + 4.33i)29-s − 3.39·31-s + ⋯
L(s)  = 1  + (0.680 + 1.17i)3-s − 1.14·5-s + (−0.695 + 1.20i)7-s + (−0.426 + 0.739i)9-s + (−0.155 − 0.269i)11-s + (−0.788 − 0.615i)13-s + (−0.779 − 1.35i)15-s + (−0.742 + 1.28i)17-s + (0.574 − 0.994i)19-s − 1.89·21-s + (0.737 + 1.27i)23-s + 0.312·25-s + 0.198·27-s + (0.464 + 0.804i)29-s − 0.609·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.923 - 0.384i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.923 - 0.384i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(416\)    =    \(2^{5} \cdot 13\)
Sign: $-0.923 - 0.384i$
Analytic conductor: \(3.32177\)
Root analytic conductor: \(1.82257\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{416} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 416,\ (\ :1/2),\ -0.923 - 0.384i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.183251 + 0.916924i\)
\(L(\frac12)\) \(\approx\) \(0.183251 + 0.916924i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 + (2.84 + 2.21i)T \)
good3 \( 1 + (-1.17 - 2.04i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + 2.56T + 5T^{2} \)
7 \( 1 + (1.84 - 3.18i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.516 + 0.895i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (3.06 - 5.30i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.50 + 4.33i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.53 - 6.12i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.5 - 4.33i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 3.39T + 31T^{2} \)
37 \( 1 + (1.06 + 1.83i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.06 - 3.57i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.19 - 7.27i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 10.7T + 47T^{2} \)
53 \( 1 + 2.56T + 53T^{2} \)
59 \( 1 + (5.23 - 9.06i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.62 + 9.73i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.86 - 8.42i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (2.50 - 4.33i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 4.31T + 73T^{2} \)
79 \( 1 - 11.5T + 79T^{2} \)
83 \( 1 + 2.64T + 83T^{2} \)
89 \( 1 + (5.34 + 9.25i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-4.90 + 8.49i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.48073350612156889841591617115, −10.66952309418903830539005993841, −9.589150915429259457327353511318, −8.980687995089332023406041062587, −8.225285020582045171652058767054, −7.11784774233260367024279026272, −5.66665981888040764719838272626, −4.63227318732739165860121632332, −3.50167979350840205087234374235, −2.80161728634387450010690245087, 0.54194731449038268215551442136, 2.40986973029628232472198183716, 3.65538940769026390032448938708, 4.69912647550065897630110722292, 6.65693849599453905043652893512, 7.24477157452063958509377813998, 7.70808897439014918392643280767, 8.789454942274541946754732718011, 9.859735895883810664936273343048, 10.89985173938690429419616128722

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