Properties

Degree 2
Conductor $ 13 \cdot 31 $
Sign $-0.419 + 0.907i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.28·2-s + (1.07 − 1.86i)3-s − 0.340·4-s + (−1.69 − 2.94i)5-s + (1.38 − 2.39i)6-s + (0.278 − 0.481i)7-s − 3.01·8-s + (−0.813 − 1.40i)9-s + (−2.18 − 3.78i)10-s + (2.38 + 4.13i)11-s + (−0.366 + 0.634i)12-s + (−0.5 − 0.866i)13-s + (0.358 − 0.620i)14-s − 7.30·15-s − 3.20·16-s + (3.25 − 5.64i)17-s + ⋯
L(s)  = 1  + 0.910·2-s + (0.620 − 1.07i)3-s − 0.170·4-s + (−0.759 − 1.31i)5-s + (0.565 − 0.979i)6-s + (0.105 − 0.182i)7-s − 1.06·8-s + (−0.271 − 0.469i)9-s + (−0.691 − 1.19i)10-s + (0.720 + 1.24i)11-s + (−0.105 + 0.183i)12-s + (−0.138 − 0.240i)13-s + (0.0957 − 0.165i)14-s − 1.88·15-s − 0.800·16-s + (0.790 − 1.36i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.419 + 0.907i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.419 + 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(403\)    =    \(13 \cdot 31\)
\( \varepsilon \)  =  $-0.419 + 0.907i$
motivic weight  =  \(1\)
character  :  $\chi_{403} (222, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 403,\ (\ :1/2),\ -0.419 + 0.907i)\)
\(L(1)\)  \(\approx\)  \(1.05626 - 1.65242i\)
\(L(\frac12)\)  \(\approx\)  \(1.05626 - 1.65242i\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{13,\;31\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{13,\;31\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad13 \( 1 + (0.5 + 0.866i)T \)
31 \( 1 + (-4.90 + 2.62i)T \)
good2 \( 1 - 1.28T + 2T^{2} \)
3 \( 1 + (-1.07 + 1.86i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.69 + 2.94i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-0.278 + 0.481i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.38 - 4.13i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-3.25 + 5.64i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.33 + 2.30i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 7.99T + 23T^{2} \)
29 \( 1 - 3.90T + 29T^{2} \)
37 \( 1 + (-5.44 + 9.43i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-6.00 - 10.4i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.19 - 7.26i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 4.74T + 47T^{2} \)
53 \( 1 + (-2.19 - 3.80i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.33 - 5.77i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + 8.01T + 61T^{2} \)
67 \( 1 + (2.05 + 3.55i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (1.84 + 3.20i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (5.46 + 9.46i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.00 - 1.74i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.62 - 8.01i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 5.05T + 89T^{2} \)
97 \( 1 + 3.19T + 97T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−11.69925065459162982793511458591, −9.711148247965847253499887148493, −9.072046420199276802184151335899, −7.931700121968334312104605217818, −7.48016489460869889387647853014, −6.10941764424272782554934576978, −4.72489564805942198576195180080, −4.29276933358678783790401090537, −2.71801235715471957602431786628, −0.994108209469360806341320947438, 2.95846333004297158195111320560, 3.71589053980853617942949715599, 4.15374233800493517698474724803, 5.73052072445235679245035291332, 6.53564161909460325864133364654, 8.104188249828714271924181435948, 8.715849850259279815534215675104, 9.960823984255752069121468182530, 10.53295921883046677770399109204, 11.75279993487884269093588544725

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