Properties

Label 2-403-13.3-c1-0-1
Degree $2$
Conductor $403$
Sign $0.951 + 0.308i$
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  + (−1.04 − 1.80i)2-s + (−0.534 − 0.926i)3-s + (−1.17 + 2.03i)4-s − 1.36·5-s + (−1.11 + 1.93i)6-s + (−2.24 + 3.88i)7-s + 0.733·8-s + (0.928 − 1.60i)9-s + (1.42 + 2.46i)10-s + (2.85 + 4.93i)11-s + 2.51·12-s + (0.209 − 3.59i)13-s + 9.35·14-s + (0.728 + 1.26i)15-s + (1.58 + 2.74i)16-s + (2.30 − 3.98i)17-s + ⋯
L(s)  = 1  + (−0.737 − 1.27i)2-s + (−0.308 − 0.534i)3-s + (−0.587 + 1.01i)4-s − 0.609·5-s + (−0.455 + 0.788i)6-s + (−0.847 + 1.46i)7-s + 0.259·8-s + (0.309 − 0.535i)9-s + (0.449 + 0.778i)10-s + (0.859 + 1.48i)11-s + 0.725·12-s + (0.0580 − 0.998i)13-s + 2.50·14-s + (0.188 + 0.325i)15-s + (0.396 + 0.686i)16-s + (0.558 − 0.966i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.951 + 0.308i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.951 + 0.308i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $0.951 + 0.308i$
Analytic conductor: \(3.21797\)
Root analytic conductor: \(1.79387\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (94, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 403,\ (\ :1/2),\ 0.951 + 0.308i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.534293 - 0.0844418i\)
\(L(\frac12)\) \(\approx\) \(0.534293 - 0.0844418i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (-0.209 + 3.59i)T \)
31 \( 1 + T \)
good2 \( 1 + (1.04 + 1.80i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (0.534 + 0.926i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + 1.36T + 5T^{2} \)
7 \( 1 + (2.24 - 3.88i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.85 - 4.93i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-2.30 + 3.98i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.32 - 2.30i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.15 - 5.45i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.07 - 7.05i)T + (-14.5 + 25.1i)T^{2} \)
37 \( 1 + (-1.33 - 2.31i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.692 - 1.20i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.24 - 3.88i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 2.65T + 47T^{2} \)
53 \( 1 - 9.07T + 53T^{2} \)
59 \( 1 + (-1.29 + 2.24i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.76 - 8.25i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.05 - 3.56i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (5.84 - 10.1i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 7.92T + 73T^{2} \)
79 \( 1 - 15.7T + 79T^{2} \)
83 \( 1 - 0.402T + 83T^{2} \)
89 \( 1 + (-7.07 - 12.2i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.50 - 9.53i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.54677596994151990794592512382, −10.14522233282176962803399574911, −9.530826446356157717829170057457, −8.896777339326662659684228198809, −7.63836377483925852203169889234, −6.61937605796152442176292639446, −5.47545832122229738831241495468, −3.72688154730830017433258288806, −2.72678489107513023915383537371, −1.31221631930230186841804004766, 0.53086246987642227830819462680, 3.63164554966607544414279555188, 4.40823652823992509088667861800, 6.01912128981656790329202755536, 6.66151319564405830706035670931, 7.54232029133086325064659045255, 8.428175486710204095067476386890, 9.321326081675306184939486898132, 10.28779082878199317759329617398, 10.95732419582609833135712402006

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