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Properties

Label 2-416-104.99-c1-0-5
Degree $2$
Conductor $416$
Sign $0.876 - 0.481i$
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  + 2.95·3-s + (−1.04 + 1.04i)5-s + (2.37 + 2.37i)7-s + 5.71·9-s + (−2.03 + 2.03i)11-s + (−3.25 − 1.55i)13-s + (−3.08 + 3.08i)15-s − 4.25i·17-s + (−0.214 − 0.214i)19-s + (7.00 + 7.00i)21-s − 0.169·23-s + 2.82i·25-s + 8.01·27-s − 9.09i·29-s + (4.96 − 4.96i)31-s + ⋯
L(s)  = 1  + 1.70·3-s + (−0.466 + 0.466i)5-s + (0.896 + 0.896i)7-s + 1.90·9-s + (−0.614 + 0.614i)11-s + (−0.902 − 0.430i)13-s + (−0.795 + 0.795i)15-s − 1.03i·17-s + (−0.0492 − 0.0492i)19-s + (1.52 + 1.52i)21-s − 0.0353·23-s + 0.564i·25-s + 1.54·27-s − 1.68i·29-s + (0.892 − 0.892i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.876 - 0.481i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.876 - 0.481i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(416\)    =    \(2^{5} \cdot 13\)
Sign: $0.876 - 0.481i$
Analytic conductor: \(3.32177\)
Root analytic conductor: \(1.82257\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{416} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 416,\ (\ :1/2),\ 0.876 - 0.481i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.19573 + 0.563687i\)
\(L(\frac12)\) \(\approx\) \(2.19573 + 0.563687i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.25 + 1.55i)T \)
good3 \( 1 - 2.95T + 3T^{2} \)
5 \( 1 + (1.04 - 1.04i)T - 5iT^{2} \)
7 \( 1 + (-2.37 - 2.37i)T + 7iT^{2} \)
11 \( 1 + (2.03 - 2.03i)T - 11iT^{2} \)
17 \( 1 + 4.25iT - 17T^{2} \)
19 \( 1 + (0.214 + 0.214i)T + 19iT^{2} \)
23 \( 1 + 0.169T + 23T^{2} \)
29 \( 1 + 9.09iT - 29T^{2} \)
31 \( 1 + (-4.96 + 4.96i)T - 31iT^{2} \)
37 \( 1 + (-2.37 - 2.37i)T + 37iT^{2} \)
41 \( 1 + (3.95 + 3.95i)T + 41iT^{2} \)
43 \( 1 + 2.48iT - 43T^{2} \)
47 \( 1 + (0.869 + 0.869i)T + 47iT^{2} \)
53 \( 1 + 1.81iT - 53T^{2} \)
59 \( 1 + (3.97 - 3.97i)T - 59iT^{2} \)
61 \( 1 + 0.851iT - 61T^{2} \)
67 \( 1 + (-8.69 - 8.69i)T + 67iT^{2} \)
71 \( 1 + (3.22 - 3.22i)T - 71iT^{2} \)
73 \( 1 + (-3.95 + 3.95i)T - 73iT^{2} \)
79 \( 1 - 10.5iT - 79T^{2} \)
83 \( 1 + (2.18 + 2.18i)T + 83iT^{2} \)
89 \( 1 + (7.97 - 7.97i)T - 89iT^{2} \)
97 \( 1 + (10.2 + 10.2i)T + 97iT^{2} \)
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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

−11.39536636452434121602358442029, −10.03029521892214285013450903954, −9.452311427207470583199716238391, −8.330740673066742317533480236868, −7.82721491020735642140184376650, −7.08811388975195452514425115371, −5.28944971660766444802435797511, −4.22515270905449091683005460553, −2.80254389216636100000346603216, −2.24725068038829795829841998935, 1.56522268946033502653853743968, 2.98501701026836102600151570476, 4.09811097164885429687007385768, 4.90831316115699694386381336803, 6.83665695936558015072856253146, 7.910419510279208246362431087752, 8.185723273702117523298059753860, 9.071484187010618442147579144928, 10.16958385158856202415439556359, 10.92860907404020328981603924488

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