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Properties

Label 2-315-9.7-c1-0-13
Degree $2$
Conductor $315$
Sign $0.939 - 0.342i$
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 + 0.866i)7-s + (1.5 − 2.59i)9-s + (3 + 1.73i)12-s + (0.5 + 0.866i)13-s + (1.5 + 0.866i)15-s + (−1.99 + 3.46i)16-s − 3·17-s + 2·19-s + (−0.999 + 1.73i)20-s + 1.73i·21-s + (−3 − 5.19i)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 + 0.327i)7-s + (0.5 − 0.866i)9-s + (0.866 + 0.499i)12-s + (0.138 + 0.240i)13-s + (0.387 + 0.223i)15-s + (−0.499 + 0.866i)16-s − 0.727·17-s + 0.458·19-s + (−0.223 + 0.387i)20-s + 0.377i·21-s + (−0.625 − 1.08i)23-s + (−0.0999 + 0.173i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.939 - 0.342i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (106, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.939 - 0.342i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.86888 + 0.329534i\)
\(L(\frac12)\) \(\approx\) \(1.86888 + 0.329534i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (-1 - 1.73i)T^{2} \)
11 \( 1 + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 3T + 17T^{2} \)
19 \( 1 - 2T + 19T^{2} \)
23 \( 1 + (3 + 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.5 + 7.79i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2 - 3.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 4T + 37T^{2} \)
41 \( 1 + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2 + 3.46i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + (3 + 5.19i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (7 - 12.1i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (7 + 12.1i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 9T + 71T^{2} \)
73 \( 1 - 11T + 73T^{2} \)
79 \( 1 + (4 - 6.92i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.5 - 7.79i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 12T + 89T^{2} \)
97 \( 1 + (1 - 1.73i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.99827690207187105144946011196, −10.85701809339300115835795985623, −9.693509170783262969859591377581, −8.665627207880056727552892041288, −7.974788613054050949889328725125, −6.90056040672647009729332444519, −6.26100813511234525757961764538, −4.24802202090610331313184639059, −3.03246470615294066701826701709, −2.14205569747094378523411173622, 1.62870338626646120767142495689, 3.07085494985228353924767551596, 4.50709596199560918723602687428, 5.54794360313809512328448145423, 6.78658440742861805794506094600, 7.85509048072689428360022057377, 9.007655204101274718009998542188, 9.745485071528360239938070055252, 10.49804384155786807780740293872, 11.35683476662989719728666577954

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