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Properties

Label 2-315-105.89-c1-0-3
Degree $2$
Conductor $315$
Sign $-0.518 - 0.855i$
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 + 1.73i)4-s + (−2.23 + 0.125i)5-s + (−1.32 + 2.29i)7-s + (−2.44 + 1.41i)11-s − 2.64·13-s + (−1.99 + 3.46i)16-s + (3.24 − 1.87i)17-s + (1.5 + 0.866i)19-s + (−2.44 − 3.74i)20-s + (−3.24 + 5.61i)23-s + (4.96 − 0.559i)25-s − 5.29·28-s + 1.41i·29-s + (4.5 − 2.59i)31-s + (2.66 − 5.28i)35-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)4-s + (−0.998 + 0.0560i)5-s + (−0.499 + 0.866i)7-s + (−0.738 + 0.426i)11-s − 0.733·13-s + (−0.499 + 0.866i)16-s + (0.785 − 0.453i)17-s + (0.344 + 0.198i)19-s + (−0.547 − 0.836i)20-s + (−0.675 + 1.17i)23-s + (0.993 − 0.111i)25-s − 0.999·28-s + 0.262i·29-s + (0.808 − 0.466i)31-s + (0.450 − 0.892i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.518 - 0.855i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.518 - 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.518 - 0.855i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ -0.518 - 0.855i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.439551 + 0.780285i\)
\(L(\frac12)\) \(\approx\) \(0.439551 + 0.780285i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.23 - 0.125i)T \)
7 \( 1 + (1.32 - 2.29i)T \)
good2 \( 1 + (-1 - 1.73i)T^{2} \)
11 \( 1 + (2.44 - 1.41i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 2.64T + 13T^{2} \)
17 \( 1 + (-3.24 + 1.87i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.5 - 0.866i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.24 - 5.61i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 1.41iT - 29T^{2} \)
31 \( 1 + (-4.5 + 2.59i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.96 - 2.29i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 4.89T + 41T^{2} \)
43 \( 1 - 4.58iT - 43T^{2} \)
47 \( 1 + (3.24 + 1.87i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.48 - 11.2i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.67 - 6.36i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (9 + 5.19i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-11.9 + 6.87i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 11.3iT - 71T^{2} \)
73 \( 1 + (6.61 + 11.4i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.5 + 6.06i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 14.9iT - 83T^{2} \)
89 \( 1 + (-8.57 + 14.8i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 5.29T + 97T^{2} \)
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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

−12.07171511953405497669759535343, −11.37387066837003019689929703684, −10.10907990046463768918753306759, −9.093702958796791432463611040285, −7.74885740228331397476315488756, −7.61808908263953923281890434228, −6.21231726568544670989697720520, −4.85813751194261999648974079246, −3.47244974783512346868170146401, −2.55856420283690482827404677128, 0.62627639910159928857147692819, 2.76505091599599628999474194035, 4.14759282954120913626160549220, 5.34106369036749191927257592183, 6.56994672163440057345520843788, 7.43444288492008158371372432639, 8.321365115855499420236913893178, 9.829947829081098610646036822489, 10.39791905127751614951769976828, 11.24180914361797925867795101041

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