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Properties

Label 2-403-31.2-c1-0-10
Degree $2$
Conductor $403$
Sign $0.867 + 0.497i$
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  + (−1.95 − 1.42i)2-s + (−0.875 + 0.636i)3-s + (1.18 + 3.65i)4-s + 2.12·5-s + 2.61·6-s + (−0.805 − 2.47i)7-s + (1.37 − 4.24i)8-s + (−0.565 + 1.73i)9-s + (−4.15 − 3.01i)10-s + (−0.213 − 0.657i)11-s + (−3.36 − 2.44i)12-s + (0.809 − 0.587i)13-s + (−1.94 + 5.99i)14-s + (−1.86 + 1.35i)15-s + (−2.49 + 1.81i)16-s + (−1.73 + 5.33i)17-s + ⋯
L(s)  = 1  + (−1.38 − 1.00i)2-s + (−0.505 + 0.367i)3-s + (0.593 + 1.82i)4-s + 0.950·5-s + 1.06·6-s + (−0.304 − 0.936i)7-s + (0.487 − 1.49i)8-s + (−0.188 + 0.579i)9-s + (−1.31 − 0.954i)10-s + (−0.0644 − 0.198i)11-s + (−0.971 − 0.705i)12-s + (0.224 − 0.163i)13-s + (−0.520 + 1.60i)14-s + (−0.480 + 0.348i)15-s + (−0.624 + 0.454i)16-s + (−0.420 + 1.29i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.867 + 0.497i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.867 + 0.497i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $0.867 + 0.497i$
Analytic conductor: \(3.21797\)
Root analytic conductor: \(1.79387\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 403,\ (\ :1/2),\ 0.867 + 0.497i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.629286 - 0.167761i\)
\(L(\frac12)\) \(\approx\) \(0.629286 - 0.167761i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (-0.809 + 0.587i)T \)
31 \( 1 + (-4.86 - 2.69i)T \)
good2 \( 1 + (1.95 + 1.42i)T + (0.618 + 1.90i)T^{2} \)
3 \( 1 + (0.875 - 0.636i)T + (0.927 - 2.85i)T^{2} \)
5 \( 1 - 2.12T + 5T^{2} \)
7 \( 1 + (0.805 + 2.47i)T + (-5.66 + 4.11i)T^{2} \)
11 \( 1 + (0.213 + 0.657i)T + (-8.89 + 6.46i)T^{2} \)
17 \( 1 + (1.73 - 5.33i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-5.93 - 4.31i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-1.66 + 5.13i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (-4.44 - 3.23i)T + (8.96 + 27.5i)T^{2} \)
37 \( 1 - 9.67T + 37T^{2} \)
41 \( 1 + (7.09 + 5.15i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + (-3.84 - 2.79i)T + (13.2 + 40.8i)T^{2} \)
47 \( 1 + (-4.56 + 3.31i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (1.04 - 3.21i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-8.16 + 5.93i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 - 7.73T + 61T^{2} \)
67 \( 1 - 7.23T + 67T^{2} \)
71 \( 1 + (3.43 - 10.5i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-0.712 - 2.19i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-0.964 + 2.96i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (1.73 + 1.25i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (-0.892 - 2.74i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-0.610 - 1.87i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.74431544968929206451641982154, −10.27554912793332319923042311280, −9.831553964750275367485719988549, −8.624671139787016885281628946114, −7.88421295503752721875217639937, −6.59492408032027763775043974778, −5.44050751356897248520840355868, −3.88851955765274147158296532659, −2.50205447601260120618860468552, −1.10517454794222535959678401783, 0.928084087597284537453788276108, 2.63497958110346024514463856565, 5.22859267967191504257229925932, 5.96280492810672291965187062478, 6.67212967842945196474743385391, 7.51310589522703140445137584892, 8.766913994666466590163223694515, 9.568654157500260183255418428074, 9.712753499849755929957462761032, 11.31496216065559016979543117047

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