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Properties

Label 2-315-105.59-c1-0-8
Degree $2$
Conductor $315$
Sign $0.596 - 0.802i$
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  + (0.956 + 1.65i)2-s + (−0.830 + 1.43i)4-s + (1.54 − 1.61i)5-s + (1.11 − 2.39i)7-s + 0.650·8-s + (4.15 + 1.00i)10-s + (2.79 + 1.61i)11-s − 4.86·13-s + (5.04 − 0.439i)14-s + (2.28 + 3.95i)16-s + (0.631 + 0.364i)17-s + (−6.81 + 3.93i)19-s + (1.04 + 3.56i)20-s + 6.17i·22-s + (−2.43 − 4.21i)23-s + ⋯
L(s)  = 1  + (0.676 + 1.17i)2-s + (−0.415 + 0.718i)4-s + (0.689 − 0.723i)5-s + (0.422 − 0.906i)7-s + 0.229·8-s + (1.31 + 0.318i)10-s + (0.842 + 0.486i)11-s − 1.34·13-s + (1.34 − 0.117i)14-s + (0.570 + 0.988i)16-s + (0.153 + 0.0884i)17-s + (−1.56 + 0.902i)19-s + (0.234 + 0.796i)20-s + 1.31i·22-s + (−0.507 − 0.879i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.596 - 0.802i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.596 - 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.596 - 0.802i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.596 - 0.802i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.89104 + 0.950324i\)
\(L(\frac12)\) \(\approx\) \(1.89104 + 0.950324i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.54 + 1.61i)T \)
7 \( 1 + (-1.11 + 2.39i)T \)
good2 \( 1 + (-0.956 - 1.65i)T + (-1 + 1.73i)T^{2} \)
11 \( 1 + (-2.79 - 1.61i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 4.86T + 13T^{2} \)
17 \( 1 + (-0.631 - 0.364i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (6.81 - 3.93i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.43 + 4.21i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 7.75iT - 29T^{2} \)
31 \( 1 + (1.23 + 0.714i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.74 + 1.58i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 1.40T + 41T^{2} \)
43 \( 1 - 6.42iT - 43T^{2} \)
47 \( 1 + (-4.21 + 2.43i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.760 + 1.31i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.15 - 5.45i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.05 - 1.18i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (9.63 + 5.56i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 10.1iT - 71T^{2} \)
73 \( 1 + (-6.91 + 11.9i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.99 + 3.44i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 4.19iT - 83T^{2} \)
89 \( 1 + (5.63 + 9.76i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 2.21T + 97T^{2} \)
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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

−12.29790007883317118719504183748, −10.67770750334351795183645471122, −9.943864180972521185029268518729, −8.722120319868496593215921571457, −7.72016933548646511957698817792, −6.81955060980517725143726351471, −5.93488023537242107289207410978, −4.74099561977296451506902605084, −4.21838679833167901726571093157, −1.75946505666453335305867922731, 2.01204395260449452813461620913, 2.74098229517502246319073965870, 4.15599667919447131101257092936, 5.32344004037930374965915743459, 6.38887814136873039469652245769, 7.66072775110806690656107412122, 9.087905703892293257264089873818, 9.886675958918217698448606835712, 10.86402536434585512584489684071, 11.59688067715526654060126260494

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