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Properties

Label 2-315-9.7-c1-0-12
Degree $2$
Conductor $315$
Sign $0.961 + 0.275i$
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  + (0.913 − 1.58i)2-s + (0.360 + 1.69i)3-s + (−0.669 − 1.15i)4-s + (0.5 + 0.866i)5-s + (3.00 + 0.977i)6-s + (0.5 − 0.866i)7-s + 1.20·8-s + (−2.74 + 1.22i)9-s + 1.82·10-s + (−0.0864 + 0.149i)11-s + (1.72 − 1.55i)12-s + (1.87 + 3.24i)13-s + (−0.913 − 1.58i)14-s + (−1.28 + 1.15i)15-s + (2.44 − 4.23i)16-s + 1.82·17-s + ⋯
L(s)  = 1  + (0.645 − 1.11i)2-s + (0.207 + 0.978i)3-s + (−0.334 − 0.579i)4-s + (0.223 + 0.387i)5-s + (1.22 + 0.399i)6-s + (0.188 − 0.327i)7-s + 0.427·8-s + (−0.913 + 0.406i)9-s + 0.577·10-s + (−0.0260 + 0.0451i)11-s + (0.497 − 0.447i)12-s + (0.519 + 0.900i)13-s + (−0.244 − 0.422i)14-s + (−0.332 + 0.299i)15-s + (0.610 − 1.05i)16-s + 0.443·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.961 + 0.275i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.961 + 0.275i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.961 + 0.275i$
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: \((2,\ 315,\ (\ :1/2),\ 0.961 + 0.275i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.05267 - 0.288484i\)
\(L(\frac12)\) \(\approx\) \(2.05267 - 0.288484i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.360 - 1.69i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (-0.913 + 1.58i)T + (-1 - 1.73i)T^{2} \)
11 \( 1 + (0.0864 - 0.149i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.87 - 3.24i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 1.82T + 17T^{2} \)
19 \( 1 + 0.138T + 19T^{2} \)
23 \( 1 + (3.90 + 6.75i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.489 + 0.847i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.36 - 2.35i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 7.88T + 37T^{2} \)
41 \( 1 + (5.02 + 8.69i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.69 - 6.40i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.90 + 5.02i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 11.7T + 53T^{2} \)
59 \( 1 + (-0.590 - 1.02i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.50 - 6.06i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.19 - 3.79i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 11.2T + 71T^{2} \)
73 \( 1 + 1.60T + 73T^{2} \)
79 \( 1 + (-7.08 + 12.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.63 + 8.03i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 11.8T + 89T^{2} \)
97 \( 1 + (5.18 - 8.98i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.57983629392882923696805870730, −10.56519957338902070278190679633, −10.29961961227173903515878945186, −9.117889961367828618341911261985, −8.009278873156317797492199308859, −6.56081865937432620482900920063, −5.12873096011274237501636197109, −4.17821570213085476392047490517, −3.32590314902189807505755629525, −2.04890801554905337656108617121, 1.60599060142200559942578632967, 3.45881547737444071391489839112, 5.15393369089035688179796604720, 5.83159034268449794056380204121, 6.70962236782138515825453260886, 7.87183131322791421179997331401, 8.264449643657793594837407473093, 9.610617883959713204255255107103, 10.94750455457095028484041910047, 12.08583272191610686072183775428

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