Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $-0.114 + 0.993i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.54 − 1.54i)2-s + (−1.73 + 0.00622i)3-s − 2.76i·4-s + (−0.252 − 2.22i)5-s + (−2.66 + 2.68i)6-s + (0.707 + 0.707i)7-s + (−1.18 − 1.18i)8-s + (2.99 − 0.0215i)9-s + (−3.82 − 3.04i)10-s + 3.38i·11-s + (0.0172 + 4.79i)12-s + (−0.206 + 0.206i)13-s + 2.18·14-s + (0.451 + 3.84i)15-s + 1.87·16-s + (0.167 − 0.167i)17-s + ⋯
L(s)  = 1  + (1.09 − 1.09i)2-s + (−0.999 + 0.00359i)3-s − 1.38i·4-s + (−0.112 − 0.993i)5-s + (−1.08 + 1.09i)6-s + (0.267 + 0.267i)7-s + (−0.419 − 0.419i)8-s + (0.999 − 0.00718i)9-s + (−1.20 − 0.961i)10-s + 1.02i·11-s + (0.00497 + 1.38i)12-s + (−0.0573 + 0.0573i)13-s + 0.583·14-s + (0.116 + 0.993i)15-s + 0.467·16-s + (0.0406 − 0.0406i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.114 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.114 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.114 + 0.993i$
motivic weight  =  \(1\)
character  :  $\chi_{105} (92, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 105,\ (\ :1/2),\ -0.114 + 0.993i)$
$L(1)$  $\approx$  $0.880395 - 0.988011i$
$L(\frac12)$  $\approx$  $0.880395 - 0.988011i$
$L(\frac{3}{2})$   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 \( 1 + (1.73 - 0.00622i)T \)
5 \( 1 + (0.252 + 2.22i)T \)
7 \( 1 + (-0.707 - 0.707i)T \)
good2 \( 1 + (-1.54 + 1.54i)T - 2iT^{2} \)
11 \( 1 - 3.38iT - 11T^{2} \)
13 \( 1 + (0.206 - 0.206i)T - 13iT^{2} \)
17 \( 1 + (-0.167 + 0.167i)T - 17iT^{2} \)
19 \( 1 - 5.31iT - 19T^{2} \)
23 \( 1 + (5.07 + 5.07i)T + 23iT^{2} \)
29 \( 1 + 2.84T + 29T^{2} \)
31 \( 1 - 9.11T + 31T^{2} \)
37 \( 1 + (5.27 + 5.27i)T + 37iT^{2} \)
41 \( 1 - 0.0314iT - 41T^{2} \)
43 \( 1 + (3.76 - 3.76i)T - 43iT^{2} \)
47 \( 1 + (3.56 - 3.56i)T - 47iT^{2} \)
53 \( 1 + (-3.55 - 3.55i)T + 53iT^{2} \)
59 \( 1 + 10.3T + 59T^{2} \)
61 \( 1 + 6.80T + 61T^{2} \)
67 \( 1 + (-6.34 - 6.34i)T + 67iT^{2} \)
71 \( 1 - 3.95iT - 71T^{2} \)
73 \( 1 + (-8.61 + 8.61i)T - 73iT^{2} \)
79 \( 1 + 11.4iT - 79T^{2} \)
83 \( 1 + (-3.88 - 3.88i)T + 83iT^{2} \)
89 \( 1 - 2.00T + 89T^{2} \)
97 \( 1 + (-2.26 - 2.26i)T + 97iT^{2} \)
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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

−13.02069986590821447137324237705, −12.14355069735409468813365864228, −12.01015647408123916409667088874, −10.61664720473131664817320712014, −9.734408443410568030401969211854, −7.938077495866738493207703938437, −6.04930299926791623817787407931, −4.92644268719996325673704644914, −4.16339290432771893384852888771, −1.76119272322542106218804930515, 3.60927862754785861988965832633, 5.00136562283711947650359555900, 6.12156746897214066136202309806, 6.88774019054993463167942204129, 7.954300532562515027183071480760, 10.08067988533238116378434941847, 11.16948577699211139955045901898, 12.00488465243977598194206009864, 13.48638273580876918637618008159, 13.89879028617906143544059529504

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