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Properties

Label 2-403-13.3-c1-0-19
Degree $2$
Conductor $403$
Sign $0.218 + 0.975i$
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  + (−0.748 − 1.29i)2-s + (0.472 + 0.819i)3-s + (−0.120 + 0.207i)4-s + 2.74·5-s + (0.707 − 1.22i)6-s + (1.47 − 2.56i)7-s − 2.63·8-s + (1.05 − 1.82i)9-s + (−2.05 − 3.55i)10-s + (2.37 + 4.11i)11-s − 0.227·12-s + (−3.20 + 1.65i)13-s − 4.42·14-s + (1.29 + 2.24i)15-s + (2.21 + 3.83i)16-s + (0.709 − 1.22i)17-s + ⋯
L(s)  = 1  + (−0.529 − 0.916i)2-s + (0.273 + 0.472i)3-s + (−0.0600 + 0.103i)4-s + 1.22·5-s + (0.288 − 0.500i)6-s + (0.559 − 0.968i)7-s − 0.931·8-s + (0.350 − 0.607i)9-s + (−0.649 − 1.12i)10-s + (0.715 + 1.23i)11-s − 0.0655·12-s + (−0.889 + 0.457i)13-s − 1.18·14-s + (0.335 + 0.580i)15-s + (0.552 + 0.957i)16-s + (0.172 − 0.298i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.218 + 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.218 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $0.218 + 0.975i$
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: \((2,\ 403,\ (\ :1/2),\ 0.218 + 0.975i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.16822 - 0.935666i\)
\(L(\frac12)\) \(\approx\) \(1.16822 - 0.935666i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (3.20 - 1.65i)T \)
31 \( 1 - T \)
good2 \( 1 + (0.748 + 1.29i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-0.472 - 0.819i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 - 2.74T + 5T^{2} \)
7 \( 1 + (-1.47 + 2.56i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.37 - 4.11i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-0.709 + 1.22i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.240 + 0.415i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.25 + 3.89i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.21 - 5.57i)T + (-14.5 + 25.1i)T^{2} \)
37 \( 1 + (5.43 + 9.40i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.05 + 7.03i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.27 - 5.67i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 1.57T + 47T^{2} \)
53 \( 1 - 3.30T + 53T^{2} \)
59 \( 1 + (1.08 - 1.87i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.31 - 12.6i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.00292 + 0.00506i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-2.21 + 3.83i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 5.00T + 73T^{2} \)
79 \( 1 + 1.66T + 79T^{2} \)
83 \( 1 - 17.6T + 83T^{2} \)
89 \( 1 + (-3.82 - 6.62i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.33 - 9.23i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.54360014418103458793503043745, −10.25875139738362071402479827969, −9.441414702906050896676998835002, −8.974281815906290363524059361896, −7.25387657360755400394832996063, −6.46549980089258793387790098390, −4.98971054630483257643987932581, −3.93159979569171766033885463391, −2.36160094497080711288517772358, −1.35312906363033445228697487990, 1.80740523362185285765111984791, 2.98757662643584554744702983023, 5.17224194170942684517680442630, 5.94402185669639256811704546572, 6.73260633437068241683901703267, 7.994522834122369810762812912289, 8.402956799351092893614030501414, 9.407389252201467263757782584449, 10.20409950183175983976596863328, 11.62973108421161134246617472756

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