Properties

Degree 2
Conductor $ 13 \cdot 31 $
Sign $0.983 + 0.178i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.14·2-s + (−0.301 − 0.521i)3-s + 2.58·4-s + (0.249 − 0.432i)5-s + (−0.644 − 1.11i)6-s + (2.01 + 3.48i)7-s + 1.25·8-s + (1.31 − 2.28i)9-s + (0.534 − 0.926i)10-s + (1.14 − 1.99i)11-s + (−0.778 − 1.34i)12-s + (−0.5 + 0.866i)13-s + (4.31 + 7.47i)14-s − 0.300·15-s − 2.48·16-s + (0.938 + 1.62i)17-s + ⋯
L(s)  = 1  + 1.51·2-s + (−0.173 − 0.301i)3-s + 1.29·4-s + (0.111 − 0.193i)5-s + (−0.263 − 0.455i)6-s + (0.761 + 1.31i)7-s + 0.442·8-s + (0.439 − 0.761i)9-s + (0.169 − 0.292i)10-s + (0.346 − 0.600i)11-s + (−0.224 − 0.389i)12-s + (−0.138 + 0.240i)13-s + (1.15 + 1.99i)14-s − 0.0776·15-s − 0.622·16-s + (0.227 + 0.394i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 + 0.178i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.983 + 0.178i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(403\)    =    \(13 \cdot 31\)
\( \varepsilon \)  =  $0.983 + 0.178i$
motivic weight  =  \(1\)
character  :  $\chi_{403} (118, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 403,\ (\ :1/2),\ 0.983 + 0.178i)\)
\(L(1)\)  \(\approx\)  \(2.99774 - 0.269360i\)
\(L(\frac12)\)  \(\approx\)  \(2.99774 - 0.269360i\)
\(L(\frac{3}{2})\)   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 \( 1 + (4.97 - 2.50i)T \)
good2 \( 1 - 2.14T + 2T^{2} \)
3 \( 1 + (0.301 + 0.521i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.249 + 0.432i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-2.01 - 3.48i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.14 + 1.99i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-0.938 - 1.62i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.05 + 3.55i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 4.23T + 23T^{2} \)
29 \( 1 - 6.47T + 29T^{2} \)
37 \( 1 + (4.34 + 7.52i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (5.02 - 8.69i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.16 + 8.94i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 5.34T + 47T^{2} \)
53 \( 1 + (4.26 - 7.39i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.60 - 9.70i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + 8.92T + 61T^{2} \)
67 \( 1 + (-3.45 + 5.97i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (5.69 - 9.86i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-5.64 + 9.78i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.487 - 0.843i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.735 + 1.27i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 0.499T + 89T^{2} \)
97 \( 1 - 8.56T + 97T^{2} \)
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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

−11.80269439180540497978264352175, −10.77577722762718655302321259112, −9.196698171625844710835882317392, −8.590802728107973615571453847609, −7.06204628561408528468449368928, −6.12629182639512999958862062949, −5.43374759039960228380525967693, −4.44905101083912471853788736537, −3.25498430113846454166743025538, −1.88763374754740957359149658232, 1.95247799464558053170035903466, 3.61566751174099258196151741607, 4.50281017210421645336176932005, 5.05023511494275241390853178346, 6.38351067440648785418223242145, 7.27051312981983816776536559982, 8.225110745629715983581773561797, 9.992550319382144839901899598578, 10.49450354381207121755903278183, 11.45817013223198799990308347715

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