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Properties

Label 2-403-13.3-c1-0-24
Degree $2$
Conductor $403$
Sign $0.681 + 0.731i$
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  + (0.578 + 1.00i)2-s + (−0.235 − 0.408i)3-s + (0.329 − 0.570i)4-s − 3.25·5-s + (0.273 − 0.473i)6-s + (0.0133 − 0.0230i)7-s + 3.07·8-s + (1.38 − 2.40i)9-s + (−1.88 − 3.26i)10-s + (−1.99 − 3.45i)11-s − 0.311·12-s + (1.93 − 3.04i)13-s + 0.0308·14-s + (0.768 + 1.33i)15-s + (1.12 + 1.94i)16-s + (−2.63 + 4.56i)17-s + ⋯
L(s)  = 1  + (0.409 + 0.709i)2-s + (−0.136 − 0.235i)3-s + (0.164 − 0.285i)4-s − 1.45·5-s + (0.111 − 0.193i)6-s + (0.00502 − 0.00871i)7-s + 1.08·8-s + (0.462 − 0.801i)9-s + (−0.596 − 1.03i)10-s + (−0.601 − 1.04i)11-s − 0.0898·12-s + (0.535 − 0.844i)13-s + 0.00823·14-s + (0.198 + 0.343i)15-s + (0.280 + 0.486i)16-s + (−0.638 + 1.10i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.681 + 0.731i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.681 + 0.731i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $0.681 + 0.731i$
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: \((2,\ 403,\ (\ :1/2),\ 0.681 + 0.731i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.22470 - 0.532653i\)
\(L(\frac12)\) \(\approx\) \(1.22470 - 0.532653i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (-1.93 + 3.04i)T \)
31 \( 1 + T \)
good2 \( 1 + (-0.578 - 1.00i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (0.235 + 0.408i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + 3.25T + 5T^{2} \)
7 \( 1 + (-0.0133 + 0.0230i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.99 + 3.45i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (2.63 - 4.56i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.34 + 7.51i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.12 - 3.68i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.84 + 4.92i)T + (-14.5 + 25.1i)T^{2} \)
37 \( 1 + (-2.46 - 4.26i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.71 + 4.70i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.26 - 5.64i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 1.95T + 47T^{2} \)
53 \( 1 - 10.7T + 53T^{2} \)
59 \( 1 + (6.64 - 11.5i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.438 - 0.760i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.37 + 4.11i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3.50 + 6.07i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 12.5T + 73T^{2} \)
79 \( 1 - 13.8T + 79T^{2} \)
83 \( 1 - 0.690T + 83T^{2} \)
89 \( 1 + (-6.83 - 11.8i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.63 - 9.76i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.19326154446881657886959833519, −10.55898646171139013073668876404, −9.100365638735125156273069434068, −7.998302384836100653921031267118, −7.38501381340719408581874430883, −6.42119863151818740542952827159, −5.48693728639884831804285233347, −4.29148249070227612526450023939, −3.24196309338890185096013649506, −0.823331380795100585917929665867, 1.97008404423175173036402610944, 3.43997007941993595868013307955, 4.30493778528031715850229974525, 5.04537223499564446179445092776, 7.12678795978258114223754948012, 7.49099773982528738531741299021, 8.484073234051149244354425507602, 9.903103424509567251020116885233, 10.77162833724493379972929455535, 11.49556274674091018743469894352

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