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Properties

Label 2-403-13.10-c1-0-13
Degree $2$
Conductor $403$
Sign $0.864 - 0.502i$
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  + (−1.98 + 1.14i)2-s + (0.827 + 1.43i)3-s + (1.61 − 2.80i)4-s − 1.16i·5-s + (−3.28 − 1.89i)6-s + (−2.58 − 1.49i)7-s + 2.82i·8-s + (0.129 − 0.223i)9-s + (1.33 + 2.31i)10-s + (−1.16 + 0.672i)11-s + 5.35·12-s + (3.15 + 1.75i)13-s + 6.84·14-s + (1.67 − 0.964i)15-s + (−0.000955 − 0.00165i)16-s + (3.33 − 5.77i)17-s + ⋯
L(s)  = 1  + (−1.40 + 0.809i)2-s + (0.477 + 0.827i)3-s + (0.809 − 1.40i)4-s − 0.521i·5-s + (−1.33 − 0.773i)6-s + (−0.978 − 0.565i)7-s + 1.00i·8-s + (0.0430 − 0.0746i)9-s + (0.421 + 0.730i)10-s + (−0.351 + 0.202i)11-s + 1.54·12-s + (0.873 + 0.486i)13-s + 1.82·14-s + (0.431 − 0.249i)15-s + (−0.000238 − 0.000413i)16-s + (0.809 − 1.40i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.864 - 0.502i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.864 - 0.502i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $0.864 - 0.502i$
Analytic conductor: \(3.21797\)
Root analytic conductor: \(1.79387\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (218, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 403,\ (\ :1/2),\ 0.864 - 0.502i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.713537 + 0.192329i\)
\(L(\frac12)\) \(\approx\) \(0.713537 + 0.192329i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 + (-3.15 - 1.75i)T \)
31 \( 1 - iT \)
good2 \( 1 + (1.98 - 1.14i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (-0.827 - 1.43i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + 1.16iT - 5T^{2} \)
7 \( 1 + (2.58 + 1.49i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.16 - 0.672i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-3.33 + 5.77i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.43 - 1.40i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.71 + 2.97i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.110 - 0.192i)T + (-14.5 + 25.1i)T^{2} \)
37 \( 1 + (-9.86 + 5.69i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (4.17 - 2.40i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.63 + 4.55i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 6.66iT - 47T^{2} \)
53 \( 1 - 4.26T + 53T^{2} \)
59 \( 1 + (1.92 + 1.11i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (7.53 - 13.0i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.16 + 2.40i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (9.09 + 5.25i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 13.9iT - 73T^{2} \)
79 \( 1 + 5.01T + 79T^{2} \)
83 \( 1 + 3.24iT - 83T^{2} \)
89 \( 1 + (-7.80 + 4.50i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.13 - 2.38i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.71475073057848168298080474787, −10.01081682691598916986152209111, −9.356943201752272719032627435240, −8.877389849772067795841192788478, −7.75891487411764892101301563438, −6.92925618296364389551447889513, −5.91524296482674106811192704125, −4.44726573440169644214271923353, −3.20907914260405785657248613219, −0.863932639230814985578196920814, 1.28253666281395980563636807502, 2.62328275593722210860914867295, 3.36494574789501896348492442191, 5.77430205344308945300060992966, 6.83389247938150759504956287305, 7.926314939218792876548654375098, 8.358372278124959655716185769699, 9.434948587309528394338282558800, 10.23298091191241681238488793599, 10.92237398017378437430578273530

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