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Properties

Label 2-403-403.216-c1-0-29
Degree $2$
Conductor $403$
Sign $-0.789 + 0.614i$
Analytic cond. $3.21797$
Root an. cond. $1.79387$
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  + (0.194 + 0.194i)2-s − 2.63i·3-s − 1.92i·4-s + (−1.36 − 1.36i)5-s + (0.512 − 0.512i)6-s + (2.99 − 2.99i)7-s + (0.764 − 0.764i)8-s − 3.93·9-s − 0.530i·10-s + (3.48 + 3.48i)11-s − 5.06·12-s + (−1.60 + 3.22i)13-s + 1.16·14-s + (−3.58 + 3.58i)15-s − 3.55·16-s − 0.413·17-s + ⋯
L(s)  = 1  + (0.137 + 0.137i)2-s − 1.52i·3-s − 0.962i·4-s + (−0.608 − 0.608i)5-s + (0.209 − 0.209i)6-s + (1.13 − 1.13i)7-s + (0.270 − 0.270i)8-s − 1.31·9-s − 0.167i·10-s + (1.05 + 1.05i)11-s − 1.46·12-s + (−0.445 + 0.895i)13-s + 0.312·14-s + (−0.925 + 0.925i)15-s − 0.887·16-s − 0.100·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.789 + 0.614i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.789 + 0.614i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $-0.789 + 0.614i$
Analytic conductor: \(3.21797\)
Root analytic conductor: \(1.79387\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (216, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 403,\ (\ :1/2),\ -0.789 + 0.614i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.492804 - 1.43550i\)
\(L(\frac12)\) \(\approx\) \(0.492804 - 1.43550i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (1.60 - 3.22i)T \)
31 \( 1 + (-5.50 - 0.851i)T \)
good2 \( 1 + (-0.194 - 0.194i)T + 2iT^{2} \)
3 \( 1 + 2.63iT - 3T^{2} \)
5 \( 1 + (1.36 + 1.36i)T + 5iT^{2} \)
7 \( 1 + (-2.99 + 2.99i)T - 7iT^{2} \)
11 \( 1 + (-3.48 - 3.48i)T + 11iT^{2} \)
17 \( 1 + 0.413T + 17T^{2} \)
19 \( 1 + (-4.50 - 4.50i)T + 19iT^{2} \)
23 \( 1 - 3.06T + 23T^{2} \)
29 \( 1 - 4.58iT - 29T^{2} \)
37 \( 1 + (7.20 + 7.20i)T + 37iT^{2} \)
41 \( 1 + (-4.32 - 4.32i)T + 41iT^{2} \)
43 \( 1 + 3.02T + 43T^{2} \)
47 \( 1 + (6.54 - 6.54i)T - 47iT^{2} \)
53 \( 1 - 9.85iT - 53T^{2} \)
59 \( 1 + (-6.57 + 6.57i)T - 59iT^{2} \)
61 \( 1 - 2.16iT - 61T^{2} \)
67 \( 1 + (6.35 + 6.35i)T + 67iT^{2} \)
71 \( 1 + (0.606 + 0.606i)T + 71iT^{2} \)
73 \( 1 + (-1.11 - 1.11i)T + 73iT^{2} \)
79 \( 1 + 10.5iT - 79T^{2} \)
83 \( 1 + (6.04 - 6.04i)T - 83iT^{2} \)
89 \( 1 + (2.14 + 2.14i)T + 89iT^{2} \)
97 \( 1 + (6.88 + 6.88i)T + 97iT^{2} \)
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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

−11.19104735842360465994470715771, −10.01730097134915863375779976826, −8.927015786425987160222403615895, −7.74094068084475596470308277292, −7.20211979548093470121355720017, −6.44401845674014091848149461790, −4.96274177458079243955311923465, −4.22884429038496946790680928016, −1.69972788532413749636415435180, −1.13674986462048635418433197126, 2.84673989929740831590024686241, 3.52386952128299101564393577738, 4.67029998791326783302192059364, 5.45560740772956567275755407384, 7.07185910130782271055870297261, 8.328296887656369872699661646615, 8.725334022926749772708712443183, 9.795145299275354385700861311228, 11.06086062062766126397075582509, 11.50721864646543697094845290257

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