Properties

Degree 2
Conductor $ 13 \cdot 31 $
Sign $0.995 - 0.0994i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.78·2-s + (−0.467 − 0.809i)3-s + 5.76·4-s + (−2.03 + 3.52i)5-s + (−1.30 − 2.25i)6-s + (−0.420 − 0.728i)7-s + 10.4·8-s + (1.06 − 1.84i)9-s + (−5.67 + 9.82i)10-s + (0.129 − 0.224i)11-s + (−2.69 − 4.67i)12-s + (−0.5 + 0.866i)13-s + (−1.17 − 2.03i)14-s + 3.80·15-s + 17.7·16-s + (−2.06 − 3.58i)17-s + ⋯
L(s)  = 1  + 1.97·2-s + (−0.269 − 0.467i)3-s + 2.88·4-s + (−0.910 + 1.57i)5-s + (−0.532 − 0.921i)6-s + (−0.159 − 0.275i)7-s + 3.71·8-s + (0.354 − 0.613i)9-s + (−1.79 + 3.10i)10-s + (0.0390 − 0.0676i)11-s + (−0.778 − 1.34i)12-s + (−0.138 + 0.240i)13-s + (−0.313 − 0.542i)14-s + 0.983·15-s + 4.43·16-s + (−0.501 − 0.869i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 - 0.0994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(403\)    =    \(13 \cdot 31\)
\( \varepsilon \)  =  $0.995 - 0.0994i$
motivic weight  =  \(1\)
character  :  $\chi_{403} (118, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 403,\ (\ :1/2),\ 0.995 - 0.0994i)\)
\(L(1)\)  \(\approx\)  \(3.57846 + 0.178397i\)
\(L(\frac12)\)  \(\approx\)  \(3.57846 + 0.178397i\)
\(L(\frac{3}{2})\)   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 \( 1 + (4.09 - 3.77i)T \)
good2 \( 1 - 2.78T + 2T^{2} \)
3 \( 1 + (0.467 + 0.809i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (2.03 - 3.52i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (0.420 + 0.728i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.129 + 0.224i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (2.06 + 3.58i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.04 - 1.81i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 4.22T + 23T^{2} \)
29 \( 1 + 8.39T + 29T^{2} \)
37 \( 1 + (-2.35 - 4.07i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.13 + 3.68i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.83 + 3.18i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 2.22T + 47T^{2} \)
53 \( 1 + (-1.29 + 2.24i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.65 - 6.32i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 - 13.7T + 61T^{2} \)
67 \( 1 + (2.02 - 3.50i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (0.900 - 1.56i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-5.05 + 8.74i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-8.05 - 13.9i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.533 - 0.923i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 0.765T + 89T^{2} \)
97 \( 1 - 0.945T + 97T^{2} \)
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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

−11.62254437357840438393287345830, −10.95001860677508509854864239045, −10.00875834974478727021235465295, −7.70388564836939498860777224292, −7.01448775116060833338926231140, −6.62504726267583734096486590660, −5.53279210093535914523152223792, −4.00136449515495669317528377404, −3.53320536874574148214781183851, −2.26910077759504164920399691564, 1.94854297802465522180152401582, 3.78851486515962175768060586376, 4.33978430220734167221617392923, 5.19542597705189264349978046501, 5.88273920911465468162527596483, 7.38174688292765123278812589575, 8.120112297689778260939643971420, 9.610170531462670796317371069733, 10.96572896283276382220650928559, 11.47315228005914679818212931641

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