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Properties

Label 2-315-105.104-c3-0-2
Degree $2$
Conductor $315$
Sign $-0.502 - 0.864i$
Analytic cond. $18.5856$
Root an. cond. $4.31110$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.826·2-s − 7.31·4-s + (−10.4 − 4.04i)5-s + (17.7 − 5.14i)7-s + 12.6·8-s + (8.61 + 3.34i)10-s − 28.3i·11-s − 29.3·13-s + (−14.7 + 4.25i)14-s + 48.0·16-s − 45.3i·17-s − 49.0i·19-s + (76.2 + 29.6i)20-s + 23.4i·22-s − 136.·23-s + ⋯
L(s)  = 1  − 0.292·2-s − 0.914·4-s + (−0.932 − 0.362i)5-s + (0.960 − 0.277i)7-s + 0.559·8-s + (0.272 + 0.105i)10-s − 0.778i·11-s − 0.627·13-s + (−0.280 + 0.0811i)14-s + 0.750·16-s − 0.646i·17-s − 0.591i·19-s + (0.852 + 0.331i)20-s + 0.227i·22-s − 1.23·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.502 - 0.864i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.502 - 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.502 - 0.864i$
Analytic conductor: \(18.5856\)
Root analytic conductor: \(4.31110\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (314, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :3/2),\ -0.502 - 0.864i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.2428210338\)
\(L(\frac12)\) \(\approx\) \(0.2428210338\)
\(L(\frac{5}{2})\) 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 \( 1 + (10.4 + 4.04i)T \)
7 \( 1 + (-17.7 + 5.14i)T \)
good2 \( 1 + 0.826T + 8T^{2} \)
11 \( 1 + 28.3iT - 1.33e3T^{2} \)
13 \( 1 + 29.3T + 2.19e3T^{2} \)
17 \( 1 + 45.3iT - 4.91e3T^{2} \)
19 \( 1 + 49.0iT - 6.85e3T^{2} \)
23 \( 1 + 136.T + 1.21e4T^{2} \)
29 \( 1 - 132. iT - 2.43e4T^{2} \)
31 \( 1 - 237. iT - 2.97e4T^{2} \)
37 \( 1 - 431. iT - 5.06e4T^{2} \)
41 \( 1 + 219.T + 6.89e4T^{2} \)
43 \( 1 - 426. iT - 7.95e4T^{2} \)
47 \( 1 + 446. iT - 1.03e5T^{2} \)
53 \( 1 + 330.T + 1.48e5T^{2} \)
59 \( 1 + 236.T + 2.05e5T^{2} \)
61 \( 1 + 574. iT - 2.26e5T^{2} \)
67 \( 1 - 104. iT - 3.00e5T^{2} \)
71 \( 1 - 814. iT - 3.57e5T^{2} \)
73 \( 1 + 292.T + 3.89e5T^{2} \)
79 \( 1 - 37.4T + 4.93e5T^{2} \)
83 \( 1 - 1.25e3iT - 5.71e5T^{2} \)
89 \( 1 - 1.40e3T + 7.04e5T^{2} \)
97 \( 1 - 107.T + 9.12e5T^{2} \)
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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

−11.54683810961082063595596855636, −10.62399607392764896938707977927, −9.560853863486475419512725058516, −8.440502898784301982325357266855, −8.110908550080151424895355757939, −6.97362099964949128808734350167, −5.14782099611525562369958053464, −4.61237366956553256741756246597, −3.35889932284014611707337475802, −1.18489497336062862513509619114, 0.11673905233283891335809438946, 1.99988981796183079857510182301, 3.92117914785178784416455515907, 4.57770299076301655724386153585, 5.85088084329778998193112186280, 7.58895901426667633099145419610, 7.88071763849232992918807818470, 8.955458980294667815328387973934, 9.993420524956472695831499543094, 10.81377299253427749670967693751

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