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Properties

Label 2-403-13.3-c1-0-3
Degree $2$
Conductor $403$
Sign $0.596 - 0.802i$
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  + (−1.36 − 2.35i)2-s + (1.42 + 2.47i)3-s + (−2.70 + 4.69i)4-s − 1.48·5-s + (3.88 − 6.72i)6-s + (1.05 − 1.81i)7-s + 9.30·8-s + (−2.56 + 4.45i)9-s + (2.01 + 3.49i)10-s + (1.86 + 3.22i)11-s − 15.4·12-s + (−3.34 − 1.35i)13-s − 5.72·14-s + (−2.11 − 3.66i)15-s + (−7.25 − 12.5i)16-s + (−1.96 + 3.40i)17-s + ⋯
L(s)  = 1  + (−0.962 − 1.66i)2-s + (0.823 + 1.42i)3-s + (−1.35 + 2.34i)4-s − 0.662·5-s + (1.58 − 2.74i)6-s + (0.397 − 0.687i)7-s + 3.28·8-s + (−0.856 + 1.48i)9-s + (0.637 + 1.10i)10-s + (0.561 + 0.973i)11-s − 4.46·12-s + (−0.927 − 0.374i)13-s − 1.52·14-s + (−0.545 − 0.945i)15-s + (−1.81 − 3.13i)16-s + (−0.476 + 0.825i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.596 - 0.802i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.596 - 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $0.596 - 0.802i$
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: \((2,\ 403,\ (\ :1/2),\ 0.596 - 0.802i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.639895 + 0.321596i\)
\(L(\frac12)\) \(\approx\) \(0.639895 + 0.321596i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (3.34 + 1.35i)T \)
31 \( 1 + T \)
good2 \( 1 + (1.36 + 2.35i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-1.42 - 2.47i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + 1.48T + 5T^{2} \)
7 \( 1 + (-1.05 + 1.81i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.86 - 3.22i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.96 - 3.40i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.97 - 5.15i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.947 - 1.64i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.39 - 5.87i)T + (-14.5 + 25.1i)T^{2} \)
37 \( 1 + (1.06 + 1.84i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.41 - 5.90i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.41 + 4.17i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 8.45T + 47T^{2} \)
53 \( 1 + 11.1T + 53T^{2} \)
59 \( 1 + (0.852 - 1.47i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.28 - 3.96i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.31 + 7.47i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-8.13 + 14.0i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 2.72T + 73T^{2} \)
79 \( 1 - 7.19T + 79T^{2} \)
83 \( 1 + 11.4T + 83T^{2} \)
89 \( 1 + (-7.10 - 12.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-9.13 + 15.8i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.85111063066140438874932681235, −10.53119153920672672271061211167, −9.724770679754950219547080782083, −9.057138232617271405174557221178, −8.131634843641458777615320669078, −7.49795258474953824727686995601, −4.58638844125758276732996147403, −4.12782111592369525666107944909, −3.27390171947377913349909025767, −1.90238517148439134017948982780, 0.59003525515615578566618867117, 2.33767416384345932360305550369, 4.58846490197042107143652870473, 5.95596089280044466836329172985, 6.81919704525854628395220729995, 7.42954029748637200023968557250, 8.271693145528398138826956875385, 8.791274865865196423319315212057, 9.474692088370968604450720967146, 11.13254595872859137669788483581

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