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Properties

Label 2-315-105.89-c1-0-15
Degree $2$
Conductor $315$
Sign $-0.961 - 0.274i$
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  + (1.28 − 2.22i)2-s + (−2.30 − 3.98i)4-s + (−2.22 − 0.260i)5-s + (−2.64 + 0.151i)7-s − 6.68·8-s + (−3.43 + 4.60i)10-s + (4.88 − 2.81i)11-s − 3.53·13-s + (−3.05 + 6.07i)14-s + (−3.98 + 6.90i)16-s + (4.62 − 2.67i)17-s + (−1.24 − 0.721i)19-s + (4.07 + 9.45i)20-s − 14.4i·22-s + (1.49 − 2.59i)23-s + ⋯
L(s)  = 1  + (0.908 − 1.57i)2-s + (−1.15 − 1.99i)4-s + (−0.993 − 0.116i)5-s + (−0.998 + 0.0573i)7-s − 2.36·8-s + (−1.08 + 1.45i)10-s + (1.47 − 0.849i)11-s − 0.980·13-s + (−0.816 + 1.62i)14-s + (−0.996 + 1.72i)16-s + (1.12 − 0.647i)17-s + (−0.286 − 0.165i)19-s + (0.910 + 2.11i)20-s − 3.08i·22-s + (0.312 − 0.541i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.961 - 0.274i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.961 - 0.274i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.961 - 0.274i$
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: \((2,\ 315,\ (\ :1/2),\ -0.961 - 0.274i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.187798 + 1.34202i\)
\(L(\frac12)\) \(\approx\) \(0.187798 + 1.34202i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.22 + 0.260i)T \)
7 \( 1 + (2.64 - 0.151i)T \)
good2 \( 1 + (-1.28 + 2.22i)T + (-1 - 1.73i)T^{2} \)
11 \( 1 + (-4.88 + 2.81i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 3.53T + 13T^{2} \)
17 \( 1 + (-4.62 + 2.67i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.24 + 0.721i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.49 + 2.59i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 2.19iT - 29T^{2} \)
31 \( 1 + (-1.31 + 0.761i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.946 - 0.546i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 6.52T + 41T^{2} \)
43 \( 1 - 0.486iT - 43T^{2} \)
47 \( 1 + (-3.68 - 2.12i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.01 - 3.48i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.70 - 9.88i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.06 - 3.50i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.15 - 1.24i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 8.13iT - 71T^{2} \)
73 \( 1 + (1.56 + 2.71i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.40 - 11.0i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 6.77iT - 83T^{2} \)
89 \( 1 + (-5.16 + 8.94i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 14.7T + 97T^{2} \)
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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

−11.60126734171137790353359878464, −10.44547441450638659255602451953, −9.601545368153231897268634073818, −8.747086086949815050974327105798, −7.10443247514315841003097779051, −5.86264111377236604030789632776, −4.57191275800003596120443038911, −3.64519704256051335838295734523, −2.83902431541130214145791485941, −0.77296301057218219405762687350, 3.42823091584136694675454885552, 4.11808778258557772132502431472, 5.28312489631025538362491650608, 6.58621402894665605668732191888, 7.04729005943332509303984385070, 7.952190316592928619478628618749, 9.037460886761431892364322617690, 10.06562175916295971853313855486, 11.85123387840735909766875396411, 12.35021515211038053509513968777

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