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Properties

Label 2-416-52.19-c1-0-5
Degree $2$
Conductor $416$
Sign $0.720 - 0.693i$
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  + (2.36 + 1.36i)3-s + (0.366 − 0.366i)5-s + (1 + 0.267i)7-s + (2.23 + 3.86i)9-s + (0.732 + 2.73i)11-s + (−3.5 − 0.866i)13-s + (1.36 − 0.366i)15-s + (−0.401 + 0.232i)17-s + (0.901 − 3.36i)19-s + (2 + 2i)21-s + (4.36 − 7.56i)23-s + 4.73i·25-s + 4.00i·27-s + (−2.23 + 3.86i)29-s + (−2.46 − 2.46i)31-s + ⋯
L(s)  = 1  + (1.36 + 0.788i)3-s + (0.163 − 0.163i)5-s + (0.377 + 0.101i)7-s + (0.744 + 1.28i)9-s + (0.220 + 0.823i)11-s + (−0.970 − 0.240i)13-s + (0.352 − 0.0945i)15-s + (−0.0974 + 0.0562i)17-s + (0.206 − 0.772i)19-s + (0.436 + 0.436i)21-s + (0.910 − 1.57i)23-s + 0.946i·25-s + 0.769i·27-s + (−0.414 + 0.717i)29-s + (−0.442 − 0.442i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.720 - 0.693i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.720 - 0.693i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(416\)    =    \(2^{5} \cdot 13\)
Sign: $0.720 - 0.693i$
Analytic conductor: \(3.32177\)
Root analytic conductor: \(1.82257\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{416} (383, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 416,\ (\ :1/2),\ 0.720 - 0.693i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.02322 + 0.815172i\)
\(L(\frac12)\) \(\approx\) \(2.02322 + 0.815172i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.5 + 0.866i)T \)
good3 \( 1 + (-2.36 - 1.36i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.366 + 0.366i)T - 5iT^{2} \)
7 \( 1 + (-1 - 0.267i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (-0.732 - 2.73i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (0.401 - 0.232i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.901 + 3.36i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-4.36 + 7.56i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.23 - 3.86i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.46 + 2.46i)T + 31iT^{2} \)
37 \( 1 + (3.59 - 0.964i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (3.06 + 11.4i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-5.73 - 9.92i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.46 - 2.46i)T - 47iT^{2} \)
53 \( 1 - 1.73T + 53T^{2} \)
59 \( 1 + (5.73 + 1.53i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-3.33 - 5.76i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-8.83 + 2.36i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-2.56 + 9.56i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (9.29 + 9.29i)T + 73iT^{2} \)
79 \( 1 - 9.46iT - 79T^{2} \)
83 \( 1 + (6.19 + 6.19i)T + 83iT^{2} \)
89 \( 1 + (-14.5 + 3.90i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (5.83 + 1.56i)T + (84.0 + 48.5i)T^{2} \)
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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

−11.08540644893882797793639824592, −10.20184144942596804761649180405, −9.306922532647257574803489682351, −8.875672475320261460091760062640, −7.77296771911421709459087765494, −6.93880239111228130495269693818, −5.13482972175239269652598510551, −4.44160399697522245671510059688, −3.14180713301880776186572677378, −2.09200528668218445280050916947, 1.56567867011703476312974804520, 2.75965147790000800075172605383, 3.79905958975421964714162278052, 5.36632675302475353950256886275, 6.70282523095941983494427964218, 7.55664661286285056521285812865, 8.251393599354806180864824494296, 9.150222578110636010331385924049, 9.933496368462688299248112237344, 11.21266396144707165182495954850

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