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Properties

Label 2-403-13.3-c1-0-26
Degree $2$
Conductor $403$
Sign $0.546 + 0.837i$
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  + (0.646 + 1.12i)2-s + (−1.49 − 2.59i)3-s + (0.163 − 0.282i)4-s + 2.87·5-s + (1.93 − 3.35i)6-s + (0.577 − 0.999i)7-s + 3.00·8-s + (−2.98 + 5.16i)9-s + (1.85 + 3.21i)10-s + (−2.03 − 3.51i)11-s − 0.976·12-s + (−2.42 + 2.66i)13-s + 1.49·14-s + (−4.29 − 7.44i)15-s + (1.62 + 2.80i)16-s + (0.246 − 0.427i)17-s + ⋯
L(s)  = 1  + (0.457 + 0.792i)2-s + (−0.864 − 1.49i)3-s + (0.0815 − 0.141i)4-s + 1.28·5-s + (0.790 − 1.36i)6-s + (0.218 − 0.377i)7-s + 1.06·8-s + (−0.993 + 1.72i)9-s + (0.587 + 1.01i)10-s + (−0.612 − 1.06i)11-s − 0.281·12-s + (−0.672 + 0.740i)13-s + 0.399·14-s + (−1.10 − 1.92i)15-s + (0.405 + 0.701i)16-s + (0.0598 − 0.103i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.546 + 0.837i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.546 + 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $0.546 + 0.837i$
Analytic conductor: \(3.21797\)
Root analytic conductor: \(1.79387\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (94, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 403,\ (\ :1/2),\ 0.546 + 0.837i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.50327 - 0.814095i\)
\(L(\frac12)\) \(\approx\) \(1.50327 - 0.814095i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (2.42 - 2.66i)T \)
31 \( 1 - T \)
good2 \( 1 + (-0.646 - 1.12i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (1.49 + 2.59i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 - 2.87T + 5T^{2} \)
7 \( 1 + (-0.577 + 0.999i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.03 + 3.51i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-0.246 + 0.427i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.72 + 2.98i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.08 - 3.61i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.28 + 2.22i)T + (-14.5 + 25.1i)T^{2} \)
37 \( 1 + (-0.599 - 1.03i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.56 + 4.44i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.00642 + 0.0111i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 9.13T + 47T^{2} \)
53 \( 1 + 9.55T + 53T^{2} \)
59 \( 1 + (7.10 - 12.3i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.60 - 9.70i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.88 + 8.46i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (5.82 - 10.0i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 14.0T + 73T^{2} \)
79 \( 1 - 9.23T + 79T^{2} \)
83 \( 1 - 6.59T + 83T^{2} \)
89 \( 1 + (-6.93 - 12.0i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3.54 + 6.14i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.15575129958547360170184294027, −10.50131411444735674211167096251, −9.228060401277280560030861842465, −7.73489586020855211750157526462, −7.15404798337817247882176798868, −6.23359293442481625018209800810, −5.69251665391789253473100675064, −4.91132372706742109576621663128, −2.35998524832538911686688827739, −1.18085591176533902496143924203, 2.12492694996255608190342785403, 3.32385766027192135829297444159, 4.76744712504138816127139999213, 5.13081882442385376768479772682, 6.22212919170003666248911832817, 7.71605712481944456270764605325, 9.238737700362542358383701371481, 10.07412887981216876628239134328, 10.38664772198458047791114322833, 11.23483662868005702490733699715

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