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Properties

Label 2-315-7.4-c1-0-0
Degree $2$
Conductor $315$
Sign $0.701 - 0.712i$
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  + (−1 − 1.73i)2-s + (−0.999 + 1.73i)4-s + (0.5 + 0.866i)5-s + (−0.5 + 2.59i)7-s + (0.999 − 1.73i)10-s + (−3 + 5.19i)11-s − 3·13-s + (5 − 1.73i)14-s + (1.99 + 3.46i)16-s + (−2 + 3.46i)17-s + (−0.5 − 0.866i)19-s − 1.99·20-s + 12·22-s + (−2 − 3.46i)23-s + (−0.499 + 0.866i)25-s + (3 + 5.19i)26-s + ⋯
L(s)  = 1  + (−0.707 − 1.22i)2-s + (−0.499 + 0.866i)4-s + (0.223 + 0.387i)5-s + (−0.188 + 0.981i)7-s + (0.316 − 0.547i)10-s + (−0.904 + 1.56i)11-s − 0.832·13-s + (1.33 − 0.462i)14-s + (0.499 + 0.866i)16-s + (−0.485 + 0.840i)17-s + (−0.114 − 0.198i)19-s − 0.447·20-s + 2.55·22-s + (−0.417 − 0.722i)23-s + (−0.0999 + 0.173i)25-s + (0.588 + 1.01i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.701 - 0.712i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.701 - 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.701 - 0.712i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (46, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.701 - 0.712i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.504297 + 0.211342i\)
\(L(\frac12)\) \(\approx\) \(0.504297 + 0.211342i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (0.5 - 2.59i)T \)
good2 \( 1 + (1 + 1.73i)T + (-1 + 1.73i)T^{2} \)
11 \( 1 + (3 - 5.19i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 3T + 13T^{2} \)
17 \( 1 + (2 - 3.46i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2 + 3.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 8T + 29T^{2} \)
31 \( 1 + (0.5 - 0.866i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.5 + 6.06i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 - T + 43T^{2} \)
47 \( 1 + (-1 - 1.73i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2 + 3.46i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4 - 6.92i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7 - 12.1i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.5 - 6.06i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + (0.5 - 0.866i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 2T + 83T^{2} \)
89 \( 1 + (6 + 10.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 6T + 97T^{2} \)
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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

−11.79936808744961153265579505286, −10.52966870836552452270170199754, −10.16459054716896308807351680399, −9.243645275053548164170171156043, −8.356856169942624795923919164453, −7.10183950851473870089493318101, −5.84022462852917182935868526168, −4.46277578097300500310241574498, −2.68432715631276831444417065643, −2.12546100867785887724192310014, 0.46840473748250614098545144252, 3.06898051867212919093631623461, 4.82233900650676937525356189528, 5.85282743592595346566181879043, 6.83674056020107552930146407063, 7.77661527540211184894592145132, 8.446575128364741982081256350388, 9.494400762207197645440963983788, 10.28993835905523746713086007712, 11.37621464236133445946525510755

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