Properties

Degree 2
Conductor $ 13 \cdot 31 $
Sign $0.412 + 0.910i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.35·2-s + (0.0933 + 0.161i)3-s − 0.160·4-s + (1.08 − 1.88i)5-s + (0.126 + 0.219i)6-s + (−1.43 − 2.49i)7-s − 2.93·8-s + (1.48 − 2.56i)9-s + (1.47 − 2.55i)10-s + (1.47 − 2.55i)11-s + (−0.0149 − 0.0259i)12-s + (−0.5 + 0.866i)13-s + (−1.95 − 3.38i)14-s + 0.406·15-s − 3.65·16-s + (0.832 + 1.44i)17-s + ⋯
L(s)  = 1  + 0.959·2-s + (0.0539 + 0.0933i)3-s − 0.0802·4-s + (0.487 − 0.843i)5-s + (0.0516 + 0.0895i)6-s + (−0.543 − 0.942i)7-s − 1.03·8-s + (0.494 − 0.855i)9-s + (0.467 − 0.808i)10-s + (0.445 − 0.770i)11-s + (−0.00432 − 0.00749i)12-s + (−0.138 + 0.240i)13-s + (−0.521 − 0.903i)14-s + 0.105·15-s − 0.913·16-s + (0.201 + 0.349i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.412 + 0.910i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.412 + 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(403\)    =    \(13 \cdot 31\)
\( \varepsilon \)  =  $0.412 + 0.910i$
motivic weight  =  \(1\)
character  :  $\chi_{403} (118, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 403,\ (\ :1/2),\ 0.412 + 0.910i)\)
\(L(1)\)  \(\approx\)  \(1.68372 - 1.08583i\)
\(L(\frac12)\)  \(\approx\)  \(1.68372 - 1.08583i\)
\(L(\frac{3}{2})\)   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 \( 1 + (4.88 + 2.66i)T \)
good2 \( 1 - 1.35T + 2T^{2} \)
3 \( 1 + (-0.0933 - 0.161i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-1.08 + 1.88i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (1.43 + 2.49i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.47 + 2.55i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-0.832 - 1.44i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.39 - 5.87i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 4.72T + 23T^{2} \)
29 \( 1 + 1.93T + 29T^{2} \)
37 \( 1 + (0.166 + 0.287i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.561 - 0.972i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.62 - 4.54i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 8.43T + 47T^{2} \)
53 \( 1 + (-5.94 + 10.2i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.23 - 5.59i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 - 5.91T + 61T^{2} \)
67 \( 1 + (5.73 - 9.93i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (0.218 - 0.377i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-2.33 + 4.04i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.84 - 11.8i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.91 - 6.77i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 4.45T + 89T^{2} \)
97 \( 1 + 1.01T + 97T^{2} \)
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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

−11.35094415136139511756288836579, −9.941614686609210642947399788072, −9.421258329306774342230708397162, −8.521923947031241840161898884865, −7.06958015599774504808832074381, −6.07195178456528597588380868478, −5.22564122735155324182882726269, −3.97657736903749057198568516328, −3.45200663145244014157335580653, −1.07438691195020510448966930793, 2.36829463064330519966885012285, 3.22946955578380940454790597860, 4.74697773939748819982877836944, 5.46611664515514538339214088747, 6.60050975300531248302495143426, 7.34534690654072487716668135194, 8.962719317101194259789700417232, 9.535364901505325667002273397389, 10.58433170996663962568295648907, 11.64627466464763148896221768157

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