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Properties

Label 2-403-403.25-c1-0-18
Degree $2$
Conductor $403$
Sign $0.717 + 0.696i$
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  − 1.46i·2-s + (0.869 + 1.50i)3-s − 0.160·4-s + (0.337 + 0.194i)5-s + (2.21 − 1.27i)6-s + (1.34 − 0.777i)7-s − 2.70i·8-s + (−0.0112 + 0.0194i)9-s + (0.286 − 0.495i)10-s + (0.846 + 0.488i)11-s + (−0.139 − 0.240i)12-s + (−2.33 + 2.75i)13-s + (−1.14 − 1.97i)14-s + 0.677i·15-s − 4.29·16-s + (1.62 + 2.81i)17-s + ⋯
L(s)  = 1  − 1.03i·2-s + (0.501 + 0.869i)3-s − 0.0800·4-s + (0.150 + 0.0870i)5-s + (0.903 − 0.521i)6-s + (0.509 − 0.293i)7-s − 0.956i·8-s + (−0.00374 + 0.00649i)9-s + (0.0905 − 0.156i)10-s + (0.255 + 0.147i)11-s + (−0.0401 − 0.0695i)12-s + (−0.646 + 0.762i)13-s + (−0.305 − 0.528i)14-s + 0.174i·15-s − 1.07·16-s + (0.393 + 0.681i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.717 + 0.696i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.717 + 0.696i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $0.717 + 0.696i$
Analytic conductor: \(3.21797\)
Root analytic conductor: \(1.79387\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 403,\ (\ :1/2),\ 0.717 + 0.696i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.76767 - 0.716343i\)
\(L(\frac12)\) \(\approx\) \(1.76767 - 0.716343i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (2.33 - 2.75i)T \)
31 \( 1 + (5.55 - 0.308i)T \)
good2 \( 1 + 1.46iT - 2T^{2} \)
3 \( 1 + (-0.869 - 1.50i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.337 - 0.194i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (-1.34 + 0.777i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.846 - 0.488i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.62 - 2.81i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.33 + 1.34i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 3.56T + 23T^{2} \)
29 \( 1 - 2.24T + 29T^{2} \)
37 \( 1 + (5.27 - 3.04i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.79 + 1.03i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.47 + 2.54i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 11.5iT - 47T^{2} \)
53 \( 1 + (-4.84 + 8.38i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.85 - 1.07i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + 10.4T + 61T^{2} \)
67 \( 1 + (10.4 + 6.00i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (5.78 + 3.34i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-9.56 - 5.52i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.68 - 4.64i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (12.9 + 7.45i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 5.51iT - 89T^{2} \)
97 \( 1 + 5.77iT - 97T^{2} \)
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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

−11.00893436758270061588817087811, −10.27218053214219339074403438913, −9.597830676194430310759441515415, −8.845703738598789153571635844559, −7.47828517023793101558031752431, −6.47856372746926199134884187200, −4.80774421085517419596479232558, −3.94707206603573765006323313394, −2.94754282629448842467027638482, −1.58028363881999483162064683132, 1.72582543993497483653397034426, 2.98271991003645717352487803298, 5.02060388727573384507163155489, 5.68637561740851920386458777138, 7.04677604764977116026519613858, 7.46668703719686679064558817994, 8.281099948224489136304263691538, 9.141419087090905075820723440480, 10.44298201755054865862307965278, 11.55659127868429522350333787448

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