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Properties

Label 2-315-315.209-c1-0-12
Degree $2$
Conductor $315$
Sign $0.761 - 0.648i$
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  + (1.32 − 1.11i)3-s + (1 + 1.73i)4-s + (−1.93 + 1.11i)5-s + (−1.32 + 2.29i)7-s + (0.5 − 2.95i)9-s + (4.81 + 2.77i)11-s + (3.25 + 1.17i)12-s + (2.24 + 3.88i)13-s + (−1.31 + 3.64i)15-s + (−1.99 + 3.46i)16-s − 7.99i·17-s + (−3.87 − 2.23i)20-s + (0.811 + 4.51i)21-s + (2.5 − 4.33i)25-s + (−2.64 − 4.47i)27-s − 5.29·28-s + ⋯
L(s)  = 1  + (0.763 − 0.645i)3-s + (0.5 + 0.866i)4-s + (−0.866 + 0.499i)5-s + (−0.499 + 0.866i)7-s + (0.166 − 0.986i)9-s + (1.45 + 0.837i)11-s + (0.940 + 0.338i)12-s + (0.622 + 1.07i)13-s + (−0.338 + 0.940i)15-s + (−0.499 + 0.866i)16-s − 1.93i·17-s + (−0.866 − 0.499i)20-s + (0.177 + 0.984i)21-s + (0.5 − 0.866i)25-s + (−0.509 − 0.860i)27-s − 0.999·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.761 - 0.648i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.761 - 0.648i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.761 - 0.648i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.761 - 0.648i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.52041 + 0.559491i\)
\(L(\frac12)\) \(\approx\) \(1.52041 + 0.559491i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.32 + 1.11i)T \)
5 \( 1 + (1.93 - 1.11i)T \)
7 \( 1 + (1.32 - 2.29i)T \)
good2 \( 1 + (-1 - 1.73i)T^{2} \)
11 \( 1 + (-4.81 - 2.77i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.24 - 3.88i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 7.99iT - 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (5.12 + 2.95i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.11 - 0.641i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 4.47iT - 71T^{2} \)
73 \( 1 - 16.9T + 73T^{2} \)
79 \( 1 + (-7.43 + 12.8i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (8.03 + 4.63i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + (1.72 - 2.98i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.83709418494622298252533757544, −11.41030650047341934365028418099, −9.418708813072473353367801003260, −8.983510627048265926429748012730, −7.80668113795538856861314300997, −6.94815893727322853010972778567, −6.50268050892042861483167850664, −4.18140609304991020353601269906, −3.27010955502497572455473284230, −2.14999361342721846717754173268, 1.25811067838583744828351335100, 3.47248947286611852944568715519, 4.03557493646433353737937445329, 5.58034703375842550976716847514, 6.68557135084563972032323604766, 7.931400307514193993636779731127, 8.745171604349723991861298413505, 9.712988530000481628060274486062, 10.76472010943604539696942993907, 11.09932224394878486617095250096

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