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Properties

Label 2-416-104.19-c1-0-6
Degree $2$
Conductor $416$
Sign $0.894 + 0.446i$
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  + (−1.05 + 1.82i)3-s + (−2.29 − 2.29i)5-s + (−1.21 − 0.324i)7-s + (−0.722 − 1.25i)9-s + (3.76 − 1.00i)11-s + (3.23 − 1.59i)13-s + (6.59 − 1.76i)15-s + (3.99 − 2.30i)17-s + (−2.52 − 0.675i)19-s + (1.86 − 1.86i)21-s + (1.62 − 2.82i)23-s + 5.49i·25-s − 3.27·27-s + (7.35 + 4.24i)29-s + (−2.29 − 2.29i)31-s + ⋯
L(s)  = 1  + (−0.608 + 1.05i)3-s + (−1.02 − 1.02i)5-s + (−0.457 − 0.122i)7-s + (−0.240 − 0.417i)9-s + (1.13 − 0.304i)11-s + (0.896 − 0.442i)13-s + (1.70 − 0.456i)15-s + (0.969 − 0.559i)17-s + (−0.578 − 0.154i)19-s + (0.407 − 0.407i)21-s + (0.339 − 0.588i)23-s + 1.09i·25-s − 0.630·27-s + (1.36 + 0.788i)29-s + (−0.411 − 0.411i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.446i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.446i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(416\)    =    \(2^{5} \cdot 13\)
Sign: $0.894 + 0.446i$
Analytic conductor: \(3.32177\)
Root analytic conductor: \(1.82257\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{416} (175, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 416,\ (\ :1/2),\ 0.894 + 0.446i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.881162 - 0.207430i\)
\(L(\frac12)\) \(\approx\) \(0.881162 - 0.207430i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.23 + 1.59i)T \)
good3 \( 1 + (1.05 - 1.82i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (2.29 + 2.29i)T + 5iT^{2} \)
7 \( 1 + (1.21 + 0.324i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (-3.76 + 1.00i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (-3.99 + 2.30i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.52 + 0.675i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-1.62 + 2.82i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-7.35 - 4.24i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.29 + 2.29i)T + 31iT^{2} \)
37 \( 1 + (1.94 + 7.25i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (0.391 + 1.46i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-9.48 + 5.47i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.31 + 3.31i)T - 47iT^{2} \)
53 \( 1 - 2.94iT - 53T^{2} \)
59 \( 1 + (0.115 - 0.432i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (9.70 - 5.60i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.48 + 9.27i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-0.740 + 2.76i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (-0.0928 - 0.0928i)T + 73iT^{2} \)
79 \( 1 - 9.86iT - 79T^{2} \)
83 \( 1 + (-7.31 + 7.31i)T - 83iT^{2} \)
89 \( 1 + (-0.299 + 0.0801i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-10.4 - 2.79i)T + (84.0 + 48.5i)T^{2} \)
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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

−11.04823725858822607369646022939, −10.41920653769642286780641991448, −9.211335582021300161942124886012, −8.694546916532034878357193294004, −7.51802109944824909244174218691, −6.20285775000030003852951985239, −5.15718228912658967021477227562, −4.21355349344332133793831872910, −3.52264078700980140143633112988, −0.75337283437286606855203255331, 1.34090913028294365264397895072, 3.20654746479519048605492654851, 4.18211051137595158519395161278, 6.11503426397286570506355882238, 6.52888165923534069801078321931, 7.36711846417605559124128374940, 8.248629509426752744620285507818, 9.507187117675753230510456349638, 10.67563056866435722994192540003, 11.49233747922562781589600924967

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