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Properties

Label 2-403-403.25-c1-0-33
Degree $2$
Conductor $403$
Sign $0.191 - 0.981i$
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  − 2.22i·2-s + (−0.613 − 1.06i)3-s − 2.97·4-s + (−2.33 − 1.34i)5-s + (−2.36 + 1.36i)6-s + (−1.24 + 0.716i)7-s + 2.16i·8-s + (0.747 − 1.29i)9-s + (−3.00 + 5.20i)10-s + (1.17 + 0.676i)11-s + (1.82 + 3.15i)12-s + (−2.61 − 2.48i)13-s + (1.59 + 2.76i)14-s + 3.31i·15-s − 1.11·16-s + (3.80 + 6.59i)17-s + ⋯
L(s)  = 1  − 1.57i·2-s + (−0.354 − 0.613i)3-s − 1.48·4-s + (−1.04 − 0.603i)5-s + (−0.967 + 0.558i)6-s + (−0.469 + 0.270i)7-s + 0.766i·8-s + (0.249 − 0.431i)9-s + (−0.950 + 1.64i)10-s + (0.353 + 0.203i)11-s + (0.526 + 0.911i)12-s + (−0.725 − 0.688i)13-s + (0.427 + 0.740i)14-s + 0.854i·15-s − 0.278·16-s + (0.923 + 1.60i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.191 - 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.191 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $0.191 - 0.981i$
Analytic conductor: \(3.21797\)
Root analytic conductor: \(1.79387\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 403,\ (\ :1/2),\ 0.191 - 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.376071 + 0.309833i\)
\(L(\frac12)\) \(\approx\) \(0.376071 + 0.309833i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (2.61 + 2.48i)T \)
31 \( 1 + (-4.29 + 3.54i)T \)
good2 \( 1 + 2.22iT - 2T^{2} \)
3 \( 1 + (0.613 + 1.06i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (2.33 + 1.34i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (1.24 - 0.716i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.17 - 0.676i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-3.80 - 6.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.731 - 0.422i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 5.80T + 23T^{2} \)
29 \( 1 + 10.0T + 29T^{2} \)
37 \( 1 + (-0.0209 + 0.0121i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (6.84 + 3.95i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.73 + 9.93i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 5.04iT - 47T^{2} \)
53 \( 1 + (-2.94 + 5.09i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (10.8 - 6.27i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 - 6.44T + 61T^{2} \)
67 \( 1 + (4.50 + 2.60i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (1.65 + 0.957i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-7.01 - 4.04i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.00 + 10.3i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-6.84 - 3.95i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 9.58iT - 89T^{2} \)
97 \( 1 + 5.23iT - 97T^{2} \)
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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

−10.71726305123679908959079882806, −9.872637549232771329126178993629, −8.973799639598386352696715616441, −7.929256997560313534712674981376, −6.85320224434430115929180328597, −5.46825695602233490092879984128, −4.07858968341653351207625381775, −3.36672337916400656462190294013, −1.68949338219158737939075885440, −0.33946952902480137315093244706, 3.27131026015550383416166681177, 4.54369235581159621198356351354, 5.21419044531160420275880191569, 6.59080268673740194432746725718, 7.27532218402281837752682928386, 7.82518081560474966682175060369, 9.177977875434478106269255903994, 9.877346252310876987066822090054, 11.18606098167963636523726283068, 11.67454947710708325391324904486

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