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Properties

Label 2-403-13.9-c1-0-33
Degree $2$
Conductor $403$
Sign $-0.980 + 0.194i$
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  + (1.06 − 1.84i)2-s + (0.793 − 1.37i)3-s + (−1.25 − 2.17i)4-s − 1.24·5-s + (−1.68 − 2.92i)6-s + (−1.62 − 2.80i)7-s − 1.09·8-s + (0.241 + 0.417i)9-s + (−1.32 + 2.29i)10-s + (1.35 − 2.35i)11-s − 3.99·12-s + (−0.214 + 3.59i)13-s − 6.89·14-s + (−0.989 + 1.71i)15-s + (1.35 − 2.33i)16-s + (−0.450 − 0.780i)17-s + ⋯
L(s)  = 1  + (0.751 − 1.30i)2-s + (0.458 − 0.793i)3-s + (−0.629 − 1.08i)4-s − 0.557·5-s + (−0.688 − 1.19i)6-s + (−0.612 − 1.06i)7-s − 0.387·8-s + (0.0804 + 0.139i)9-s + (−0.419 + 0.726i)10-s + (0.409 − 0.709i)11-s − 1.15·12-s + (−0.0594 + 0.998i)13-s − 1.84·14-s + (−0.255 + 0.442i)15-s + (0.337 − 0.584i)16-s + (−0.109 − 0.189i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.980 + 0.194i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.980 + 0.194i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $-0.980 + 0.194i$
Analytic conductor: \(3.21797\)
Root analytic conductor: \(1.79387\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (373, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 403,\ (\ :1/2),\ -0.980 + 0.194i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.198889 - 2.02385i\)
\(L(\frac12)\) \(\approx\) \(0.198889 - 2.02385i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (0.214 - 3.59i)T \)
31 \( 1 - T \)
good2 \( 1 + (-1.06 + 1.84i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-0.793 + 1.37i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + 1.24T + 5T^{2} \)
7 \( 1 + (1.62 + 2.80i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.35 + 2.35i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (0.450 + 0.780i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.39 - 2.41i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.13 - 5.43i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.78 + 4.82i)T + (-14.5 - 25.1i)T^{2} \)
37 \( 1 + (-6.01 + 10.4i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.0645 + 0.111i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.65 - 6.33i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 12.7T + 47T^{2} \)
53 \( 1 + 6.98T + 53T^{2} \)
59 \( 1 + (-1.68 - 2.91i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.96 - 3.40i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.66 - 13.2i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.57 + 4.46i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 4.79T + 73T^{2} \)
79 \( 1 + 8.45T + 79T^{2} \)
83 \( 1 - 13.8T + 83T^{2} \)
89 \( 1 + (5.17 - 8.96i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.76 + 4.78i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.14203685655028305116751666974, −10.17620032183945352823675166547, −9.297635854530942937886711024849, −7.82903018360153647600357831395, −7.27314386807568049832554628375, −5.96465237500445252238689771694, −4.24700983437055703920908304054, −3.76442186070004651159534947279, −2.45635105039965770856744755614, −1.12004588634987449998715242148, 2.92384178103331695803474796446, 4.04616636758577910283723005831, 4.87680402465785875544949430476, 5.99691355812262634438001460630, 6.81569904580132914097562817619, 7.933855418030992726519896770928, 8.746318804895494259112888776602, 9.656309422081324586957251003873, 10.59494132262262982812078506586, 12.17402026568069111332149481750

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