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Properties

Label 2-315-63.20-c1-0-9
Degree $2$
Conductor $315$
Sign $0.858 - 0.513i$
Analytic cond. $2.51528$
Root an. cond. $1.58596$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.5 + 0.866i)3-s + (−1 − 1.73i)4-s + (0.5 + 0.866i)5-s + (0.5 + 2.59i)7-s + (1.5 + 2.59i)9-s + (1.5 + 0.866i)11-s − 3.46i·12-s + 1.73i·15-s + (−1.99 + 3.46i)16-s + 6·17-s − 6.92i·19-s + (0.999 − 1.73i)20-s + (−1.5 + 4.33i)21-s + (3 − 1.73i)23-s + (−0.499 + 0.866i)25-s + ⋯
L(s)  = 1  + (0.866 + 0.499i)3-s + (−0.5 − 0.866i)4-s + (0.223 + 0.387i)5-s + (0.188 + 0.981i)7-s + (0.5 + 0.866i)9-s + (0.452 + 0.261i)11-s − 0.999i·12-s + 0.447i·15-s + (−0.499 + 0.866i)16-s + 1.45·17-s − 1.58i·19-s + (0.223 − 0.387i)20-s + (−0.327 + 0.944i)21-s + (0.625 − 0.361i)23-s + (−0.0999 + 0.173i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.858 - 0.513i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.858 - 0.513i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.858 - 0.513i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (146, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.858 - 0.513i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.60796 + 0.444317i\)
\(L(\frac12)\) \(\approx\) \(1.60796 + 0.444317i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.5 - 0.866i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (-0.5 - 2.59i)T \)
good2 \( 1 + (1 + 1.73i)T^{2} \)
11 \( 1 + (-1.5 - 0.866i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 6T + 17T^{2} \)
19 \( 1 + 6.92iT - 19T^{2} \)
23 \( 1 + (-3 + 1.73i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (7.5 + 4.33i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (3 - 1.73i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 8T + 37T^{2} \)
41 \( 1 + (-3 - 5.19i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (5 - 8.66i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.5 + 7.79i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 13.8iT - 53T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3 + 1.73i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (4 + 6.92i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 1.73iT - 71T^{2} \)
73 \( 1 + 8.66iT - 73T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.5 - 7.79i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 + (1.5 + 0.866i)T + (48.5 + 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.54940125514033349827971486924, −10.62289719620489971099593457558, −9.594715708349409728341693737168, −9.218239396363103385386719090672, −8.211679878006017215844390951679, −6.89603850950307380551989556456, −5.56724679533352150436560579419, −4.76327946629470109463104142599, −3.28810225659307748280111801935, −1.91833119246151998377989065887, 1.41554289810592598254256106855, 3.38748318244313275216838997956, 3.98710211753925211293857233408, 5.59173401334293336786224068185, 7.24462315635382303326744935270, 7.69282245746074592107972122002, 8.698210855951129210896653710099, 9.447018375853162589119407153150, 10.49624122302047485102978478565, 11.98891165242698916670482941680

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