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Properties

Label 2-403-13.4-c1-0-21
Degree $2$
Conductor $403$
Sign $0.441 + 0.897i$
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  + (0.720 + 0.415i)2-s + (−1.53 + 2.65i)3-s + (−0.653 − 1.13i)4-s − 1.49i·5-s + (−2.20 + 1.27i)6-s + (1.75 − 1.01i)7-s − 2.75i·8-s + (−3.19 − 5.53i)9-s + (0.622 − 1.07i)10-s + (−4.84 − 2.79i)11-s + 4.00·12-s + (−3.59 − 0.284i)13-s + 1.68·14-s + (3.97 + 2.29i)15-s + (−0.163 + 0.282i)16-s + (0.509 + 0.881i)17-s + ⋯
L(s)  = 1  + (0.509 + 0.294i)2-s + (−0.884 + 1.53i)3-s + (−0.326 − 0.566i)4-s − 0.669i·5-s + (−0.901 + 0.520i)6-s + (0.661 − 0.381i)7-s − 0.972i·8-s + (−1.06 − 1.84i)9-s + (0.197 − 0.341i)10-s + (−1.46 − 0.843i)11-s + 1.15·12-s + (−0.996 − 0.0789i)13-s + 0.449·14-s + (1.02 + 0.592i)15-s + (−0.0407 + 0.0706i)16-s + (0.123 + 0.213i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.441 + 0.897i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.441 + 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $0.441 + 0.897i$
Analytic conductor: \(3.21797\)
Root analytic conductor: \(1.79387\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (342, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 403,\ (\ :1/2),\ 0.441 + 0.897i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.673732 - 0.419329i\)
\(L(\frac12)\) \(\approx\) \(0.673732 - 0.419329i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (3.59 + 0.284i)T \)
31 \( 1 - iT \)
good2 \( 1 + (-0.720 - 0.415i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (1.53 - 2.65i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + 1.49iT - 5T^{2} \)
7 \( 1 + (-1.75 + 1.01i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (4.84 + 2.79i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-0.509 - 0.881i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-5.64 + 3.25i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.35 + 2.34i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.95 - 3.38i)T + (-14.5 - 25.1i)T^{2} \)
37 \( 1 + (9.35 + 5.40i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.13 + 0.652i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.725 + 1.25i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 8.81iT - 47T^{2} \)
53 \( 1 + 3.69T + 53T^{2} \)
59 \( 1 + (-1.50 + 0.869i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.78 - 10.0i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.51 - 1.45i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (3.05 - 1.76i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 11.6iT - 73T^{2} \)
79 \( 1 - 14.5T + 79T^{2} \)
83 \( 1 + 15.3iT - 83T^{2} \)
89 \( 1 + (-0.250 - 0.144i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (9.34 - 5.39i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.73863966062568131634640009674, −10.45536991790993298142558953021, −9.460239415494416988594893403174, −8.632096561314733496367353302739, −7.16801111573943381500010814667, −5.63151058930634449336382097140, −5.09391230354485416673118362392, −4.74743630430973405511943643086, −3.40391521592017943572212353539, −0.48199426253242342806675569036, 1.98739913078394706195657516194, 2.93531599165580247540233043095, 5.01984742917056200251267834688, 5.37483992938876141826258638270, 6.89389315248817425335562597867, 7.65754377357081898414646171163, 8.081833703946063823777053928166, 9.817333167300369029577589616879, 11.00529181271970910457054103221, 11.70131411449823642871601340955

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