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Properties

Label 2-403-403.36-c1-0-21
Degree $2$
Conductor $403$
Sign $-0.313 + 0.949i$
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  + (−1.90 + 1.10i)2-s − 1.41·3-s + (1.42 − 2.47i)4-s + (1.04 − 0.603i)5-s + (2.69 − 1.55i)6-s + (0.354 − 0.204i)7-s + 1.88i·8-s − 1.00·9-s + (−1.32 + 2.30i)10-s + (3.29 + 1.90i)11-s + (−2.01 + 3.49i)12-s + (−3.33 − 1.37i)13-s + (−0.450 + 0.779i)14-s + (−1.47 + 0.852i)15-s + (0.781 + 1.35i)16-s + (−0.765 − 1.32i)17-s + ⋯
L(s)  = 1  + (−1.34 + 0.778i)2-s − 0.815·3-s + (0.713 − 1.23i)4-s + (0.467 − 0.269i)5-s + (1.09 − 0.635i)6-s + (0.133 − 0.0772i)7-s + 0.665i·8-s − 0.335·9-s + (−0.420 + 0.728i)10-s + (0.994 + 0.574i)11-s + (−0.581 + 1.00i)12-s + (−0.924 − 0.382i)13-s + (−0.120 + 0.208i)14-s + (−0.381 + 0.220i)15-s + (0.195 + 0.338i)16-s + (−0.185 − 0.321i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.313 + 0.949i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.313 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $-0.313 + 0.949i$
Analytic conductor: \(3.21797\)
Root analytic conductor: \(1.79387\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (36, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 403,\ (\ :1/2),\ -0.313 + 0.949i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0811390 - 0.112274i\)
\(L(\frac12)\) \(\approx\) \(0.0811390 - 0.112274i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (3.33 + 1.37i)T \)
31 \( 1 + (-4.92 + 2.60i)T \)
good2 \( 1 + (1.90 - 1.10i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + 1.41T + 3T^{2} \)
5 \( 1 + (-1.04 + 0.603i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (-0.354 + 0.204i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-3.29 - 1.90i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (0.765 + 1.32i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (6.57 - 3.79i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.09 + 3.62i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.41 + 5.91i)T + (-14.5 + 25.1i)T^{2} \)
37 \( 1 + 2.62iT - 37T^{2} \)
41 \( 1 + (4.05 + 2.34i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.53 + 7.86i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 13.0iT - 47T^{2} \)
53 \( 1 + (-3.49 - 6.06i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.53 - 2.61i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.35 - 5.80i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (10.3 + 5.99i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 13.5iT - 71T^{2} \)
73 \( 1 + (7.85 - 4.53i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.23 - 7.34i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.33 - 0.769i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (3.18 - 1.84i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (7.94 + 4.58i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.57543112974829211588455124762, −10.00725659126562331374072895665, −9.102979845347821714531338430656, −8.304769220633153651555685143788, −7.25849384126856710685310299684, −6.35382815538858230291627955415, −5.66711367776105094061451713263, −4.30952775631737981812342421303, −1.92536183185580290073814024564, −0.15134252729065496769903447111, 1.63298553369151755828996630108, 2.92927418492539477932506677089, 4.70462998120449250161740704589, 6.07334404691022659068989113812, 6.85011073178780744981084786472, 8.245430419414520312586289804963, 8.957486135505031953216912710097, 9.837240978494074315751590484467, 10.62443748066263614792642033706, 11.42721427257966397982071420843

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