Properties

Degree 2
Conductor $ 3^{2} \cdot 5 \cdot 7 $
Sign $-0.389 - 0.921i$
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 + 4.93i·4-s + (1.50 − 1.65i)5-s + (−2.20 + 1.46i)7-s + (−5.45 + 5.45i)8-s + (5.88 − 0.272i)10-s + 1.46·11-s + (0.887 + 0.887i)13-s + (−6.82 − 1.38i)14-s − 10.4·16-s + (2.10 − 2.10i)17-s − 3.95·19-s + (8.14 + 7.42i)20-s + (2.72 + 2.72i)22-s + (4.13 − 4.13i)23-s + ⋯
L(s)  = 1  + (1.31 + 1.31i)2-s + 2.46i·4-s + (0.673 − 0.739i)5-s + (−0.833 + 0.552i)7-s + (−1.92 + 1.92i)8-s + (1.85 − 0.0862i)10-s + 0.441·11-s + (0.246 + 0.246i)13-s + (−1.82 − 0.370i)14-s − 2.61·16-s + (0.510 − 0.510i)17-s − 0.908·19-s + (1.82 + 1.66i)20-s + (0.580 + 0.580i)22-s + (0.861 − 0.861i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.389 - 0.921i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.389 - 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.389 - 0.921i$
motivic weight  =  \(1\)
character  :  $\chi_{315} (307, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 315,\ (\ :1/2),\ -0.389 - 0.921i)\)
\(L(1)\)  \(\approx\)  \(1.40722 + 2.12195i\)
\(L(\frac12)\)  \(\approx\)  \(1.40722 + 2.12195i\)
\(L(\frac{3}{2})\)   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 \( 1 \)
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

−12.46413260848507226258258337649, −11.53040104387717600493464255505, −9.754868213420316560901617363863, −8.875995510808256735613482103061, −7.975466920996786691908041850994, −6.56224523465579132447809464018, −6.15323485766459842403972481650, −5.09726497252649659121005129982, −4.14107984936719408165601052816, −2.72500554779121630459903588014, 1.59761248262855392885765881437, 3.08484462030732182228073319763, 3.72498179260205712718079700406, 5.17121459246462828069240184329, 6.18516145679623492061507675880, 6.96756877096130075449469885137, 9.050268452293682295224463817048, 10.10936782464110333793882548296, 10.52426539346745650986371391413, 11.37326511937204177433600466043

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