Properties

Degree 2
Conductor $ 3^{2} \cdot 5 \cdot 7 $
Sign $-0.717 + 0.696i$
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 + (1.46 + 2.20i)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 + (0.552 + 0.833i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.717 + 0.696i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.717 + 0.696i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.717 + 0.696i$
motivic weight  =  \(1\)
character  :  $\chi_{315} (118, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 315,\ (\ :1/2),\ -0.717 + 0.696i)\)
\(L(1)\)  \(\approx\)  \(0.878940 - 2.16719i\)
\(L(\frac12)\)  \(\approx\)  \(0.878940 - 2.16719i\)
\(L(\frac{3}{2})\)   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 \( 1 + (-1.46 - 2.20i)T \)
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

−11.62883041820573325471117756241, −10.95514344546520241162736515193, −9.522277061107717951100677761012, −8.904741489254686335807316630306, −7.33581964943913813714774306507, −5.72077563233768575787658125317, −4.98029228481635652516381955080, −4.08612448238047100063171536641, −2.83687800586061885755554196393, −1.38595822707340274749855971369, 3.07369463881918122498341207968, 4.10666712069369793881875330038, 4.90794204553580621524577965629, 6.28224009133700522274188870184, 7.06322505034408472492430055027, 7.72815979836781930681085516741, 8.634917267862497602099855869755, 10.38469006458678977926192617377, 11.43246456528422862224123091383, 12.15868866907023746669087525967

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