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Properties

Label 2-315-35.4-c1-0-1
Degree $2$
Conductor $315$
Sign $-0.464 - 0.885i$
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  + (−2.05 + 1.18i)2-s + (1.82 − 3.15i)4-s + (−1.46 − 1.69i)5-s + (−2.03 − 1.68i)7-s + 3.91i·8-s + (5.01 + 1.74i)10-s + (−1.80 + 3.13i)11-s + 3.36i·13-s + (6.19 + 1.04i)14-s + (−1 − 1.73i)16-s + (4.71 + 2.72i)17-s + (2.32 + 4.02i)19-s + (−8.00 + 1.53i)20-s − 8.58i·22-s + (0.599 − 0.346i)23-s + ⋯
L(s)  = 1  + (−1.45 + 0.840i)2-s + (0.911 − 1.57i)4-s + (−0.654 − 0.756i)5-s + (−0.771 − 0.636i)7-s + 1.38i·8-s + (1.58 + 0.551i)10-s + (−0.544 + 0.943i)11-s + 0.934i·13-s + (1.65 + 0.278i)14-s + (−0.250 − 0.433i)16-s + (1.14 + 0.660i)17-s + (0.532 + 0.923i)19-s + (−1.79 + 0.343i)20-s − 1.83i·22-s + (0.125 − 0.0722i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.464 - 0.885i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.464 - 0.885i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.464 - 0.885i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ -0.464 - 0.885i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.190377 + 0.314747i\)
\(L(\frac12)\) \(\approx\) \(0.190377 + 0.314747i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (1.46 + 1.69i)T \)
7 \( 1 + (2.03 + 1.68i)T \)
good2 \( 1 + (2.05 - 1.18i)T + (1 - 1.73i)T^{2} \)
11 \( 1 + (1.80 - 3.13i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 3.36iT - 13T^{2} \)
17 \( 1 + (-4.71 - 2.72i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.32 - 4.02i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.599 + 0.346i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 8.00T + 29T^{2} \)
31 \( 1 + (2.5 - 4.33i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.51 - 2.60i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 5.17T + 41T^{2} \)
43 \( 1 + 8.91iT - 43T^{2} \)
47 \( 1 + (3.38 - 1.95i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.57 - 3.21i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.388 - 0.673i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.82 + 4.88i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-11.9 - 6.90i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 8.00T + 71T^{2} \)
73 \( 1 + (-1.31 - 0.760i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.96 + 8.60i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 3.76iT - 83T^{2} \)
89 \( 1 + (-1.80 - 3.13i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 12.2iT - 97T^{2} \)
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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

−11.97229111972057136180885942347, −10.42834196478944198875264262243, −10.00127921588827876971260389144, −9.025636706617690736586205491140, −8.140299073549854570636574415773, −7.37165644493543247050480608149, −6.59450039652226026915359435899, −5.22614080898871675195994496223, −3.77099246138375557846825064093, −1.33309854967637266458344140583, 0.46823908691287377438200036522, 2.84253646839756830314169316886, 3.19044814396910905797211177545, 5.50067586712485086053189919824, 6.90744372913908199539472554837, 7.88265649680507682425471325398, 8.574717987643125062477832550646, 9.689774667010752851131285157424, 10.32122269155449870980446364054, 11.21836083077413917330081898908

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