Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $0.376 + 0.926i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.86 + 1.86i)2-s + (−0.707 + 0.707i)3-s − 4.93i·4-s + (−1.50 − 1.65i)5-s − 2.63i·6-s + (−2.20 − 1.46i)7-s + (5.45 + 5.45i)8-s − 1.00i·9-s + (5.88 + 0.272i)10-s − 1.46·11-s + (3.48 + 3.48i)12-s + (0.887 − 0.887i)13-s + (6.82 − 1.38i)14-s + (2.23 + 0.103i)15-s − 10.4·16-s + (−2.10 − 2.10i)17-s + ⋯
L(s)  = 1  + (−1.31 + 1.31i)2-s + (−0.408 + 0.408i)3-s − 2.46i·4-s + (−0.673 − 0.739i)5-s − 1.07i·6-s + (−0.833 − 0.552i)7-s + (1.92 + 1.92i)8-s − 0.333i·9-s + (1.85 + 0.0862i)10-s − 0.441·11-s + (1.00 + 1.00i)12-s + (0.246 − 0.246i)13-s + (1.82 − 0.370i)14-s + (0.576 + 0.0267i)15-s − 2.61·16-s + (−0.510 − 0.510i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.376 + 0.926i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.376 + 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.376 + 0.926i$
motivic weight  =  \(1\)
character  :  $\chi_{105} (13, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 105,\ (\ :1/2),\ 0.376 + 0.926i)\)
\(L(1)\)  \(\approx\)  \(0.135051 - 0.0909174i\)
\(L(\frac12)\)  \(\approx\)  \(0.135051 - 0.0909174i\)
\(L(\frac{3}{2})\)   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 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 + (1.50 + 1.65i)T \)
7 \( 1 + (2.20 + 1.46i)T \)
good2 \( 1 + (1.86 - 1.86i)T - 2iT^{2} \)
11 \( 1 + 1.46T + 11T^{2} \)
13 \( 1 + (-0.887 + 0.887i)T - 13iT^{2} \)
17 \( 1 + (2.10 + 2.10i)T + 17iT^{2} \)
19 \( 1 + 3.95T + 19T^{2} \)
23 \( 1 + (4.13 + 4.13i)T + 23iT^{2} \)
29 \( 1 + 5.18iT - 29T^{2} \)
31 \( 1 - 6.10iT - 31T^{2} \)
37 \( 1 + (-2.25 + 2.25i)T - 37iT^{2} \)
41 \( 1 + 0.769iT - 41T^{2} \)
43 \( 1 + (5.18 + 5.18i)T + 43iT^{2} \)
47 \( 1 + (-8.57 - 8.57i)T + 47iT^{2} \)
53 \( 1 + (0.544 + 0.544i)T + 53iT^{2} \)
59 \( 1 + 3.19T + 59T^{2} \)
61 \( 1 + 1.42iT - 61T^{2} \)
67 \( 1 + (5.93 - 5.93i)T - 67iT^{2} \)
71 \( 1 - 7.62T + 71T^{2} \)
73 \( 1 + (-6.81 + 6.81i)T - 73iT^{2} \)
79 \( 1 + 4.52iT - 79T^{2} \)
83 \( 1 + (6.75 - 6.75i)T - 83iT^{2} \)
89 \( 1 - 1.19T + 89T^{2} \)
97 \( 1 + (8.68 + 8.68i)T + 97iT^{2} \)
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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

−13.88269766636085984973145874675, −12.53425832087928521586305170788, −10.94029947286595669524606411986, −10.09208572868992456349048746359, −9.050191379174690464192890435030, −8.127187284594634838192887162522, −6.98356448749066069374825438976, −5.88595552663908538024157028993, −4.45467361731555604987258853525, −0.28372214435530800514007010656, 2.33618419782065677740164138893, 3.73752906561084477145288517939, 6.44135385567376657214164644855, 7.68649335478160699898695876944, 8.697202527044341030044839306205, 9.941397902590740758591154989417, 10.82713150845164531869135554608, 11.63913310242691888831359367982, 12.46936200013936938340720892113, 13.35823354181017488348315899364

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