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Properties

Label 2-315-15.8-c1-0-1
Degree $2$
Conductor $315$
Sign $0.670 - 0.741i$
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  + (−0.876 − 0.876i)2-s − 0.464i·4-s + (−0.0251 + 2.23i)5-s + (−0.707 + 0.707i)7-s + (−2.15 + 2.15i)8-s + (1.98 − 1.93i)10-s + 4.66i·11-s + (1.54 + 1.54i)13-s + 1.23·14-s + 2.85·16-s + (−2.32 − 2.32i)17-s + 3.54i·19-s + (1.03 + 0.0117i)20-s + (4.08 − 4.08i)22-s + (−1.44 + 1.44i)23-s + ⋯
L(s)  = 1  + (−0.619 − 0.619i)2-s − 0.232i·4-s + (−0.0112 + 0.999i)5-s + (−0.267 + 0.267i)7-s + (−0.763 + 0.763i)8-s + (0.626 − 0.612i)10-s + 1.40i·11-s + (0.429 + 0.429i)13-s + 0.331·14-s + 0.713·16-s + (−0.565 − 0.565i)17-s + 0.813i·19-s + (0.232 + 0.00261i)20-s + (0.870 − 0.870i)22-s + (−0.300 + 0.300i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.670 - 0.741i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.670 - 0.741i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.670 - 0.741i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.670 - 0.741i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.679568 + 0.301862i\)
\(L(\frac12)\) \(\approx\) \(0.679568 + 0.301862i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.0251 - 2.23i)T \)
7 \( 1 + (0.707 - 0.707i)T \)
good2 \( 1 + (0.876 + 0.876i)T + 2iT^{2} \)
11 \( 1 - 4.66iT - 11T^{2} \)
13 \( 1 + (-1.54 - 1.54i)T + 13iT^{2} \)
17 \( 1 + (2.32 + 2.32i)T + 17iT^{2} \)
19 \( 1 - 3.54iT - 19T^{2} \)
23 \( 1 + (1.44 - 1.44i)T - 23iT^{2} \)
29 \( 1 - 7.58T + 29T^{2} \)
31 \( 1 - 8.12T + 31T^{2} \)
37 \( 1 + (6.00 - 6.00i)T - 37iT^{2} \)
41 \( 1 + 6.72iT - 41T^{2} \)
43 \( 1 + (-5.46 - 5.46i)T + 43iT^{2} \)
47 \( 1 + (5.16 + 5.16i)T + 47iT^{2} \)
53 \( 1 + (6.86 - 6.86i)T - 53iT^{2} \)
59 \( 1 - 6.40T + 59T^{2} \)
61 \( 1 + 13.6T + 61T^{2} \)
67 \( 1 + (-2.77 + 2.77i)T - 67iT^{2} \)
71 \( 1 + 0.576iT - 71T^{2} \)
73 \( 1 + (4.94 + 4.94i)T + 73iT^{2} \)
79 \( 1 - 2.50iT - 79T^{2} \)
83 \( 1 + (-3.99 + 3.99i)T - 83iT^{2} \)
89 \( 1 + 6.86T + 89T^{2} \)
97 \( 1 + (-4.43 + 4.43i)T - 97iT^{2} \)
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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

−11.73654270735696150323919233630, −10.61428322725988319957986914569, −10.04711182535584919339503929833, −9.308822798347371536710122314131, −8.148136565689072827347845291593, −6.89116905940535880418652464154, −6.09973752275417134167553095549, −4.61753242489219466200912094235, −2.98251398896435187621302298518, −1.86029513879543281851376968163, 0.66436259923093449259876878491, 3.14342578009131277519658723100, 4.40966519576649831770878825668, 5.88129337739383766956763883142, 6.68987397809248319082376812941, 8.112057008921433800508028385156, 8.480187085312822218220324010343, 9.308996293121970636856668168949, 10.46147839115460317584860910965, 11.56118076799169476239761610762

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