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Properties

Label 2-403-31.25-c1-0-7
Degree $2$
Conductor $403$
Sign $-0.913 - 0.407i$
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  − 0.899·2-s + (1.46 + 2.54i)3-s − 1.19·4-s + (−1.03 + 1.79i)5-s + (−1.31 − 2.28i)6-s + (1.98 + 3.42i)7-s + 2.87·8-s + (−2.80 + 4.85i)9-s + (0.929 − 1.61i)10-s + (0.890 − 1.54i)11-s + (−1.74 − 3.02i)12-s + (0.5 − 0.866i)13-s + (−1.78 − 3.08i)14-s − 6.06·15-s − 0.200·16-s + (−1.36 − 2.36i)17-s + ⋯
L(s)  = 1  − 0.636·2-s + (0.846 + 1.46i)3-s − 0.595·4-s + (−0.462 + 0.800i)5-s + (−0.538 − 0.933i)6-s + (0.748 + 1.29i)7-s + 1.01·8-s + (−0.934 + 1.61i)9-s + (0.294 − 0.509i)10-s + (0.268 − 0.464i)11-s + (−0.504 − 0.873i)12-s + (0.138 − 0.240i)13-s + (−0.476 − 0.824i)14-s − 1.56·15-s − 0.0502·16-s + (−0.331 − 0.574i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.913 - 0.407i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.913 - 0.407i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $-0.913 - 0.407i$
Analytic conductor: \(3.21797\)
Root analytic conductor: \(1.79387\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (118, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 403,\ (\ :1/2),\ -0.913 - 0.407i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.216475 + 1.01753i\)
\(L(\frac12)\) \(\approx\) \(0.216475 + 1.01753i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 + (5.42 - 1.25i)T \)
good2 \( 1 + 0.899T + 2T^{2} \)
3 \( 1 + (-1.46 - 2.54i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (1.03 - 1.79i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-1.98 - 3.42i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.890 + 1.54i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.36 + 2.36i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.362 + 0.627i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 3.30T + 23T^{2} \)
29 \( 1 - 7.78T + 29T^{2} \)
37 \( 1 + (4.34 + 7.51i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-5.71 + 9.90i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.18 - 8.98i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 1.30T + 47T^{2} \)
53 \( 1 + (2.69 - 4.67i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.50 - 4.34i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 - 6.62T + 61T^{2} \)
67 \( 1 + (4.90 - 8.49i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-4.01 + 6.94i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-0.246 + 0.427i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.43 - 4.21i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.21 + 3.83i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 13.8T + 89T^{2} \)
97 \( 1 + 13.7T + 97T^{2} \)
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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

−11.11288468388456384409434662142, −10.68816867666341115268964682621, −9.610810624001269222442328965990, −8.894390434196997980217159314985, −8.462370961293996712318946670641, −7.46845324905028903351146678278, −5.62815646623464128739630938330, −4.64646756396353484811369766176, −3.65218621922053345825604739802, −2.49307698141362561594089349101, 0.844073978655704109206207626335, 1.76463779969776789837648445982, 3.87613701505731408826686202091, 4.72435486971877562088991906057, 6.61686323584477092220176861608, 7.51417778557812571206258445339, 8.200208077727672959488700842882, 8.580001061725042963930973938306, 9.711464924254069143444254006467, 10.78540264766820792117713197857

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