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Properties

Label 2-315-5.4-c3-0-15
Degree $2$
Conductor $315$
Sign $-0.925 - 0.379i$
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  + 5.18i·2-s − 18.9·4-s + (4.24 − 10.3i)5-s − 7i·7-s − 56.5i·8-s + (53.6 + 22.0i)10-s + 35.9·11-s + 45.2i·13-s + 36.3·14-s + 142.·16-s + 113. i·17-s − 61.5·19-s + (−80.2 + 195. i)20-s + 186. i·22-s + 30.6i·23-s + ⋯
L(s)  = 1  + 1.83i·2-s − 2.36·4-s + (0.379 − 0.925i)5-s − 0.377i·7-s − 2.49i·8-s + (1.69 + 0.695i)10-s + 0.985·11-s + 0.965i·13-s + 0.693·14-s + 2.21·16-s + 1.61i·17-s − 0.743·19-s + (−0.896 + 2.18i)20-s + 1.80i·22-s + 0.277i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.925 - 0.379i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.925 - 0.379i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.925 - 0.379i$
Analytic conductor: \(18.5856\)
Root analytic conductor: \(4.31110\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :3/2),\ -0.925 - 0.379i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.569857943\)
\(L(\frac12)\) \(\approx\) \(1.569857943\)
\(L(\frac{5}{2})\) 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 \( 1 + (-4.24 + 10.3i)T \)
7 \( 1 + 7iT \)
good2 \( 1 - 5.18iT - 8T^{2} \)
11 \( 1 - 35.9T + 1.33e3T^{2} \)
13 \( 1 - 45.2iT - 2.19e3T^{2} \)
17 \( 1 - 113. iT - 4.91e3T^{2} \)
19 \( 1 + 61.5T + 6.85e3T^{2} \)
23 \( 1 - 30.6iT - 1.21e4T^{2} \)
29 \( 1 - 214.T + 2.43e4T^{2} \)
31 \( 1 - 164.T + 2.97e4T^{2} \)
37 \( 1 - 410. iT - 5.06e4T^{2} \)
41 \( 1 - 309.T + 6.89e4T^{2} \)
43 \( 1 + 29.9iT - 7.95e4T^{2} \)
47 \( 1 - 483. iT - 1.03e5T^{2} \)
53 \( 1 - 295. iT - 1.48e5T^{2} \)
59 \( 1 - 416.T + 2.05e5T^{2} \)
61 \( 1 + 151.T + 2.26e5T^{2} \)
67 \( 1 + 89.5iT - 3.00e5T^{2} \)
71 \( 1 + 714.T + 3.57e5T^{2} \)
73 \( 1 + 1.13e3iT - 3.89e5T^{2} \)
79 \( 1 - 323.T + 4.93e5T^{2} \)
83 \( 1 + 297. iT - 5.71e5T^{2} \)
89 \( 1 + 90.2T + 7.04e5T^{2} \)
97 \( 1 - 492. iT - 9.12e5T^{2} \)
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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

−11.93569024997445960179900673012, −10.28816799506027805359560457288, −9.270020128035885840094252932713, −8.619371813209324943763221473306, −7.85004356369732468541665544863, −6.43449015632295105147147409882, −6.20307731972487931834991710847, −4.67720381074770329618010822740, −4.15116786836363813530151561968, −1.24674431491320195155520504337, 0.68179947778149013736599678988, 2.29105408978410771753540740339, 3.02950032572610545216058115148, 4.23430480804563317238657321098, 5.53484709661316556428102863094, 6.89415123557170804755574899558, 8.425664878038062928503306923865, 9.387361033284448631154820897762, 10.09171634666016739920770594829, 10.88316080471710167025854940182

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