Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $-0.994 + 0.107i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.800 + 0.800i)2-s + (−1.34 + 1.09i)3-s + 0.718i·4-s + (−2.10 − 0.754i)5-s + (0.199 − 1.95i)6-s + (−0.707 − 0.707i)7-s + (−2.17 − 2.17i)8-s + (0.606 − 2.93i)9-s + (2.28 − 1.08i)10-s + 5.20i·11-s + (−0.785 − 0.964i)12-s + (−3.24 + 3.24i)13-s + 1.13·14-s + (3.65 − 1.28i)15-s + 2.04·16-s + (−0.844 + 0.844i)17-s + ⋯
L(s)  = 1  + (−0.566 + 0.566i)2-s + (−0.775 + 0.631i)3-s + 0.359i·4-s + (−0.941 − 0.337i)5-s + (0.0813 − 0.796i)6-s + (−0.267 − 0.267i)7-s + (−0.769 − 0.769i)8-s + (0.202 − 0.979i)9-s + (0.723 − 0.341i)10-s + 1.56i·11-s + (−0.226 − 0.278i)12-s + (−0.900 + 0.900i)13-s + 0.302·14-s + (0.942 − 0.332i)15-s + 0.511·16-s + (−0.204 + 0.204i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.107i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 + 0.107i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.994 + 0.107i$
motivic weight  =  \(1\)
character  :  $\chi_{105} (92, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 105,\ (\ :1/2),\ -0.994 + 0.107i)$
$L(1)$  $\approx$  $0.0164494 - 0.305890i$
$L(\frac12)$  $\approx$  $0.0164494 - 0.305890i$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 + (1.34 - 1.09i)T \)
5 \( 1 + (2.10 + 0.754i)T \)
7 \( 1 + (0.707 + 0.707i)T \)
good2 \( 1 + (0.800 - 0.800i)T - 2iT^{2} \)
11 \( 1 - 5.20iT - 11T^{2} \)
13 \( 1 + (3.24 - 3.24i)T - 13iT^{2} \)
17 \( 1 + (0.844 - 0.844i)T - 17iT^{2} \)
19 \( 1 + 1.32iT - 19T^{2} \)
23 \( 1 + (-5.62 - 5.62i)T + 23iT^{2} \)
29 \( 1 + 4.38T + 29T^{2} \)
31 \( 1 + 1.70T + 31T^{2} \)
37 \( 1 + (1.71 + 1.71i)T + 37iT^{2} \)
41 \( 1 - 1.82iT - 41T^{2} \)
43 \( 1 + (0.281 - 0.281i)T - 43iT^{2} \)
47 \( 1 + (-3.39 + 3.39i)T - 47iT^{2} \)
53 \( 1 + (3.51 + 3.51i)T + 53iT^{2} \)
59 \( 1 + 1.81T + 59T^{2} \)
61 \( 1 + 2.47T + 61T^{2} \)
67 \( 1 + (-7.92 - 7.92i)T + 67iT^{2} \)
71 \( 1 + 9.06iT - 71T^{2} \)
73 \( 1 + (1.33 - 1.33i)T - 73iT^{2} \)
79 \( 1 - 11.5iT - 79T^{2} \)
83 \( 1 + (-5.46 - 5.46i)T + 83iT^{2} \)
89 \( 1 - 9.43T + 89T^{2} \)
97 \( 1 + (3.06 + 3.06i)T + 97iT^{2} \)
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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

−14.95647120427447497965298220436, −12.86377445425472077496137289238, −12.16882945141633340304598509630, −11.25625128031783369867165019823, −9.737253694109208089725560303036, −9.083349072964505764401357060177, −7.42755700220672470297617779739, −6.88902580504292156197687281097, −4.90925692396880439146783094245, −3.81562744118819017688671602845, 0.45003944395854031226311914675, 2.87468705437903145394906106039, 5.21771769028195638957121827458, 6.37648541752969062603149652887, 7.76106909985536843117266630747, 8.873586870297137036854743816626, 10.48340622640896363659995435522, 11.04107560668326730116545774900, 11.92440654516512049811214747475, 12.86490800344784559249076205292

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