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Properties

Label 2-315-5.4-c3-0-29
Degree $2$
Conductor $315$
Sign $0.144 + 0.989i$
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  − 3.18i·2-s − 2.13·4-s + (11.0 − 1.61i)5-s + 7i·7-s − 18.6i·8-s + (−5.14 − 35.2i)10-s + 68.7·11-s + 56.2i·13-s + 22.2·14-s − 76.5·16-s + 37.9i·17-s + 26.0·19-s + (−23.5 + 3.44i)20-s − 218. i·22-s − 25.5i·23-s + ⋯
L(s)  = 1  − 1.12i·2-s − 0.266·4-s + (0.989 − 0.144i)5-s + 0.377i·7-s − 0.825i·8-s + (−0.162 − 1.11i)10-s + 1.88·11-s + 1.19i·13-s + 0.425·14-s − 1.19·16-s + 0.541i·17-s + 0.314·19-s + (−0.263 + 0.0385i)20-s − 2.11i·22-s − 0.231i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.144 + 0.989i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.144 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.144 + 0.989i$
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: \((2,\ 315,\ (\ :3/2),\ 0.144 + 0.989i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.691982513\)
\(L(\frac12)\) \(\approx\) \(2.691982513\)
\(L(\frac{5}{2})\) 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 \( 1 + (-11.0 + 1.61i)T \)
7 \( 1 - 7iT \)
good2 \( 1 + 3.18iT - 8T^{2} \)
11 \( 1 - 68.7T + 1.33e3T^{2} \)
13 \( 1 - 56.2iT - 2.19e3T^{2} \)
17 \( 1 - 37.9iT - 4.91e3T^{2} \)
19 \( 1 - 26.0T + 6.85e3T^{2} \)
23 \( 1 + 25.5iT - 1.21e4T^{2} \)
29 \( 1 - 148.T + 2.43e4T^{2} \)
31 \( 1 - 75.7T + 2.97e4T^{2} \)
37 \( 1 + 120. iT - 5.06e4T^{2} \)
41 \( 1 + 345.T + 6.89e4T^{2} \)
43 \( 1 - 287. iT - 7.95e4T^{2} \)
47 \( 1 + 528. iT - 1.03e5T^{2} \)
53 \( 1 + 361. iT - 1.48e5T^{2} \)
59 \( 1 + 705.T + 2.05e5T^{2} \)
61 \( 1 + 393.T + 2.26e5T^{2} \)
67 \( 1 + 591. iT - 3.00e5T^{2} \)
71 \( 1 - 668.T + 3.57e5T^{2} \)
73 \( 1 + 251. iT - 3.89e5T^{2} \)
79 \( 1 + 295.T + 4.93e5T^{2} \)
83 \( 1 - 916. iT - 5.71e5T^{2} \)
89 \( 1 + 736.T + 7.04e5T^{2} \)
97 \( 1 - 142. iT - 9.12e5T^{2} \)
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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

−11.15780102485457524305062476713, −10.05090374204279264327544510362, −9.388225072172285518241956823526, −8.694680966894307862825162091379, −6.74307207960102053170599640654, −6.30010375640370419991964425501, −4.64291258636543128807000832443, −3.50383719630697865595925879255, −2.07304177561899430584012268601, −1.28285969056331516908710949401, 1.32345601131585768116879138074, 3.01210228711498818720051750400, 4.70130914010067052201027809285, 5.82855727054274273792963242671, 6.53313109052782793340625742694, 7.34010982399783022135711460733, 8.506930386287574238707803854681, 9.399244772249097763919932080459, 10.35714288839712996543016220050, 11.42063684910465394917329298832

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