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Properties

Label 2-403-31.25-c1-0-10
Degree $2$
Conductor $403$
Sign $-0.0776 - 0.996i$
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  − 2.75·2-s + (1.04 + 1.81i)3-s + 5.58·4-s + (1.03 − 1.79i)5-s + (−2.88 − 4.99i)6-s + (2.05 + 3.56i)7-s − 9.87·8-s + (−0.696 + 1.20i)9-s + (−2.85 + 4.94i)10-s + (−2.89 + 5.02i)11-s + (5.85 + 10.1i)12-s + (0.5 − 0.866i)13-s + (−5.66 − 9.81i)14-s + 4.34·15-s + 16.0·16-s + (0.558 + 0.967i)17-s + ⋯
L(s)  = 1  − 1.94·2-s + (0.605 + 1.04i)3-s + 2.79·4-s + (0.463 − 0.802i)5-s + (−1.17 − 2.04i)6-s + (0.777 + 1.34i)7-s − 3.49·8-s + (−0.232 + 0.402i)9-s + (−0.902 + 1.56i)10-s + (−0.873 + 1.51i)11-s + (1.68 + 2.92i)12-s + (0.138 − 0.240i)13-s + (−1.51 − 2.62i)14-s + 1.12·15-s + 4.00·16-s + (0.135 + 0.234i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0776 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0776 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $-0.0776 - 0.996i$
Analytic conductor: \(3.21797\)
Root analytic conductor: \(1.79387\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (118, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 403,\ (\ :1/2),\ -0.0776 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.533840 + 0.577055i\)
\(L(\frac12)\) \(\approx\) \(0.533840 + 0.577055i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 + (3.68 + 4.17i)T \)
good2 \( 1 + 2.75T + 2T^{2} \)
3 \( 1 + (-1.04 - 1.81i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-1.03 + 1.79i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-2.05 - 3.56i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.89 - 5.02i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-0.558 - 0.967i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.17 + 2.03i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 1.07T + 23T^{2} \)
29 \( 1 + 1.28T + 29T^{2} \)
37 \( 1 + (-1.12 - 1.94i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.14 - 1.97i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.75 - 3.04i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 9.81T + 47T^{2} \)
53 \( 1 + (-0.394 + 0.682i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.600 - 1.04i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + 14.8T + 61T^{2} \)
67 \( 1 + (3.85 - 6.67i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-6.95 + 12.0i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-0.390 + 0.677i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.613 + 1.06i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.22 + 2.12i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 17.3T + 89T^{2} \)
97 \( 1 - 1.70T + 97T^{2} \)
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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

−11.00959025167042901141134059954, −10.20162901494771984410399773990, −9.385250156176052248668492570465, −9.037885120266270945997662829405, −8.261249396442217749569610909384, −7.39755731776312583227562995110, −5.84201385165681013436042781134, −4.80047639177557607247547385777, −2.71668508664027578130810082970, −1.78608572335992420620562517808, 0.914633002965215516114715252655, 2.09353287308981013106589385327, 3.18500207606418960175491719637, 5.91725869072415719833395405705, 6.96853125583755923866757095796, 7.49476844773293404120855575370, 8.187308574982014641654910016028, 8.915893974279575700430178674430, 10.32918789720971601041309501528, 10.68212493036062313517202377626

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