# Properties

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

# Related objects

## 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 : $\chi_{403} (222, \cdot )$ 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

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