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Properties

Label 2-403-13.10-c1-0-19
Degree $2$
Conductor $403$
Sign $0.766 + 0.642i$
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  + (−0.117 + 0.0679i)2-s + (0.0351 + 0.0608i)3-s + (−0.990 + 1.71i)4-s − 1.52i·5-s + (−0.00827 − 0.00477i)6-s + (−2.01 − 1.16i)7-s − 0.541i·8-s + (1.49 − 2.59i)9-s + (0.103 + 0.179i)10-s + (3.90 − 2.25i)11-s − 0.139·12-s + (−0.579 + 3.55i)13-s + 0.316·14-s + (0.0925 − 0.0534i)15-s + (−1.94 − 3.36i)16-s + (3.84 − 6.65i)17-s + ⋯
L(s)  = 1  + (−0.0832 + 0.0480i)2-s + (0.0202 + 0.0351i)3-s + (−0.495 + 0.858i)4-s − 0.680i·5-s + (−0.00337 − 0.00195i)6-s + (−0.761 − 0.439i)7-s − 0.191i·8-s + (0.499 − 0.864i)9-s + (0.0326 + 0.0566i)10-s + (1.17 − 0.678i)11-s − 0.0402·12-s + (−0.160 + 0.986i)13-s + 0.0845·14-s + (0.0239 − 0.0138i)15-s + (−0.486 − 0.842i)16-s + (0.932 − 1.61i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $0.766 + 0.642i$
Analytic conductor: \(3.21797\)
Root analytic conductor: \(1.79387\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (218, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 403,\ (\ :1/2),\ 0.766 + 0.642i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.07502 - 0.391139i\)
\(L(\frac12)\) \(\approx\) \(1.07502 - 0.391139i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (0.579 - 3.55i)T \)
31 \( 1 - iT \)
good2 \( 1 + (0.117 - 0.0679i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (-0.0351 - 0.0608i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + 1.52iT - 5T^{2} \)
7 \( 1 + (2.01 + 1.16i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-3.90 + 2.25i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-3.84 + 6.65i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.14 - 1.23i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.33 - 4.03i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.95 + 5.12i)T + (-14.5 + 25.1i)T^{2} \)
37 \( 1 + (9.11 - 5.26i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.07 + 1.20i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.71 + 9.89i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 4.21iT - 47T^{2} \)
53 \( 1 + 9.42T + 53T^{2} \)
59 \( 1 + (-0.350 - 0.202i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.09 + 7.10i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.21 - 3.01i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-9.29 - 5.36i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 11.9iT - 73T^{2} \)
79 \( 1 + 4.61T + 79T^{2} \)
83 \( 1 - 6.34iT - 83T^{2} \)
89 \( 1 + (12.9 - 7.48i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-13.3 - 7.68i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.56936855466172281797037560713, −9.724370653535678949057659596025, −9.397479894096133074228952563146, −8.651920159240387879116821278829, −7.28884635813637187018607184322, −6.73058727086718135207051185462, −5.20635000441487723304376679258, −3.95540189005317869582273215883, −3.34845681855638781341034793876, −0.888937018901528940714795530173, 1.57198521336762766006721176044, 3.17833232667896230588708526847, 4.54053373138028258227925855946, 5.69355292646500650970667405787, 6.53877709721584040832754475785, 7.58177936408540616011139187695, 8.825821138372908383455328208138, 9.683776183370460179043702146017, 10.43783524777267933500431021815, 10.97574342110415206953589019458

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