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Properties

Label 2-403-13.3-c1-0-16
Degree $2$
Conductor $403$
Sign $0.594 - 0.804i$
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  + (−0.0361 − 0.0626i)2-s + (1.64 + 2.84i)3-s + (0.997 − 1.72i)4-s + 2.88·5-s + (0.119 − 0.206i)6-s + (−0.540 + 0.936i)7-s − 0.289·8-s + (−3.91 + 6.78i)9-s + (−0.104 − 0.180i)10-s + (−0.248 − 0.430i)11-s + 6.56·12-s + (−3.34 − 1.34i)13-s + 0.0782·14-s + (4.74 + 8.22i)15-s + (−1.98 − 3.43i)16-s + (1.78 − 3.08i)17-s + ⋯
L(s)  = 1  + (−0.0255 − 0.0443i)2-s + (0.949 + 1.64i)3-s + (0.498 − 0.863i)4-s + 1.29·5-s + (0.0486 − 0.0842i)6-s + (−0.204 + 0.354i)7-s − 0.102·8-s + (−1.30 + 2.26i)9-s + (−0.0330 − 0.0571i)10-s + (−0.0749 − 0.129i)11-s + 1.89·12-s + (−0.928 − 0.371i)13-s + 0.0209·14-s + (1.22 + 2.12i)15-s + (−0.496 − 0.859i)16-s + (0.431 − 0.747i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.594 - 0.804i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.594 - 0.804i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $0.594 - 0.804i$
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: \((2,\ 403,\ (\ :1/2),\ 0.594 - 0.804i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.98751 + 1.00267i\)
\(L(\frac12)\) \(\approx\) \(1.98751 + 1.00267i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (3.34 + 1.34i)T \)
31 \( 1 + T \)
good2 \( 1 + (0.0361 + 0.0626i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-1.64 - 2.84i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 - 2.88T + 5T^{2} \)
7 \( 1 + (0.540 - 0.936i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.248 + 0.430i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.78 + 3.08i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.404 + 0.700i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.65 + 6.33i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.83 - 3.18i)T + (-14.5 + 25.1i)T^{2} \)
37 \( 1 + (-4.90 - 8.49i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.601 + 1.04i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.71 + 2.96i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 6.86T + 47T^{2} \)
53 \( 1 + 2.70T + 53T^{2} \)
59 \( 1 + (6.99 - 12.1i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.98 + 10.3i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.53 + 6.12i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.98 + 3.43i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 3.80T + 73T^{2} \)
79 \( 1 - 11.9T + 79T^{2} \)
83 \( 1 - 9.74T + 83T^{2} \)
89 \( 1 + (-4.47 - 7.75i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (7.17 - 12.4i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.83519689977265855739102581403, −10.25832436187791547563982450539, −9.623469727347279021163684190917, −9.208616425645525672510673344802, −7.972956597360467769620607164139, −6.39302688704154468530259824462, −5.36217218631788101540921105872, −4.75412626621600385729419137085, −2.96513117026180492651026096007, −2.27428266567601787829996387495, 1.74071048044807492298198705863, 2.45325101788930156337043028357, 3.64930744885260331176222810452, 5.85357044402291611515644982509, 6.61077732229817196869832478633, 7.49666523036878005185048236868, 8.035713545755033986067789749277, 9.185415865581142529828958100207, 9.925492358271314510712793828743, 11.47987488299815181612132793120

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