Properties

Degree 2
Conductor $ 13 \cdot 31 $
Sign $-0.341 + 0.940i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.680·2-s + (0.896 − 1.55i)3-s − 1.53·4-s + (−0.204 − 0.354i)5-s + (0.609 − 1.05i)6-s + (1.47 − 2.55i)7-s − 2.40·8-s + (−0.106 − 0.183i)9-s + (−0.139 − 0.240i)10-s + (−2.64 − 4.58i)11-s + (−1.37 + 2.38i)12-s + (−0.5 − 0.866i)13-s + (1.00 − 1.73i)14-s − 0.732·15-s + 1.43·16-s + (−3.14 + 5.44i)17-s + ⋯
L(s)  = 1  + 0.481·2-s + (0.517 − 0.896i)3-s − 0.768·4-s + (−0.0914 − 0.158i)5-s + (0.248 − 0.431i)6-s + (0.557 − 0.965i)7-s − 0.850·8-s + (−0.0353 − 0.0612i)9-s + (−0.0439 − 0.0761i)10-s + (−0.797 − 1.38i)11-s + (−0.397 + 0.688i)12-s + (−0.138 − 0.240i)13-s + (0.268 − 0.464i)14-s − 0.189·15-s + 0.359·16-s + (−0.762 + 1.32i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.341 + 0.940i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.341 + 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(403\)    =    \(13 \cdot 31\)
\( \varepsilon \)  =  $-0.341 + 0.940i$
motivic weight  =  \(1\)
character  :  $\chi_{403} (222, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 403,\ (\ :1/2),\ -0.341 + 0.940i)\)
\(L(1)\)  \(\approx\)  \(0.881954 - 1.25828i\)
\(L(\frac12)\)  \(\approx\)  \(0.881954 - 1.25828i\)
\(L(\frac{3}{2})\)   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 \( 1 + (-4.66 + 3.03i)T \)
good2 \( 1 - 0.680T + 2T^{2} \)
3 \( 1 + (-0.896 + 1.55i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.204 + 0.354i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-1.47 + 2.55i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.64 + 4.58i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (3.14 - 5.44i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.61 + 2.80i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 1.03T + 23T^{2} \)
29 \( 1 + 0.536T + 29T^{2} \)
37 \( 1 + (2.42 - 4.19i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (4.29 + 7.43i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.91 - 3.32i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 5.01T + 47T^{2} \)
53 \( 1 + (-7.09 - 12.2i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.839 - 1.45i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 - 0.829T + 61T^{2} \)
67 \( 1 + (-2.70 - 4.68i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3.75 - 6.51i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-6.20 - 10.7i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.37 + 7.57i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (5.05 + 8.75i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 17.4T + 89T^{2} \)
97 \( 1 - 12.7T + 97T^{2} \)
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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

−10.94134525350241327683652254567, −10.26447739518114754255656375766, −8.651551114415496814970568775814, −8.348314134873545463093035242806, −7.39252624057235628483620935081, −6.18697016807823819018615614138, −5.00861893845425010007033692355, −4.02021014497346668267916786953, −2.72651244742714235345760125168, −0.870217187233968550440883429473, 2.45413258453439915323322296069, 3.63705193508388435586988667839, 4.97818221441786902754014908487, 5.02711824484864403949363929418, 6.82389492993937908444713613691, 8.073443640949921462925901899469, 9.060636984570484928781769647877, 9.536293594728480621684844428456, 10.41489416047798807585156527292, 11.75248477131341797720899314863

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