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Properties

Label 2-416-13.8-c0-0-0
Degree $2$
Conductor $416$
Sign $0.881 - 0.471i$
Analytic cond. $0.207611$
Root an. cond. $0.455643$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 + i)5-s − 9-s + i·13-s − 2i·17-s + i·25-s + (1 − i)37-s + (−1 − i)41-s + (−1 − i)45-s + i·49-s − 2·53-s − 2·61-s + (−1 + i)65-s + (1 − i)73-s + 81-s + (2 − 2i)85-s + ⋯
L(s)  = 1  + (1 + i)5-s − 9-s + i·13-s − 2i·17-s + i·25-s + (1 − i)37-s + (−1 − i)41-s + (−1 − i)45-s + i·49-s − 2·53-s − 2·61-s + (−1 + i)65-s + (1 − i)73-s + 81-s + (2 − 2i)85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 - 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 - 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(416\)    =    \(2^{5} \cdot 13\)
Sign: $0.881 - 0.471i$
Analytic conductor: \(0.207611\)
Root analytic conductor: \(0.455643\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{416} (385, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 416,\ (\ :0),\ 0.881 - 0.471i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9133529951\)
\(L(\frac12)\) \(\approx\) \(0.9133529951\)
\(L(1)\) 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 \( 1 - iT \)
good3 \( 1 + T^{2} \)
5 \( 1 + (-1 - i)T + iT^{2} \)
7 \( 1 - iT^{2} \)
11 \( 1 - iT^{2} \)
17 \( 1 + 2iT - T^{2} \)
19 \( 1 + iT^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + iT^{2} \)
37 \( 1 + (-1 + i)T - iT^{2} \)
41 \( 1 + (1 + i)T + iT^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + 2T + T^{2} \)
59 \( 1 - iT^{2} \)
61 \( 1 + 2T + T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 + iT^{2} \)
73 \( 1 + (-1 + i)T - iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + (1 - i)T - iT^{2} \)
97 \( 1 + (-1 - i)T + iT^{2} \)
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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

−11.35159825788314744739280946884, −10.72184922602421536327586719363, −9.522346323367501895718308396668, −9.133629002780230953314977305340, −7.66184863277729996560368845822, −6.71394351084427805407794498246, −5.93769959301795202666695387774, −4.84197831786097185235208915102, −3.15044123081260147708106111956, −2.23985557201118057159815943019, 1.63117168589590306032425342219, 3.16866821116746938307111895430, 4.71912387984632983114618621555, 5.72025115057794121051462155357, 6.27893573026064318113826885391, 8.078498275041504390228503343954, 8.511196947928718310135508737938, 9.566869835608252480345014322619, 10.35723704972721346096764210950, 11.31069611276589590934868735387

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