Properties

Label 2-315-63.25-c1-0-24
Degree $2$
Conductor $315$
Sign $-0.937 + 0.348i$
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  − 1.61·2-s + (0.678 − 1.59i)3-s + 0.593·4-s + (0.5 − 0.866i)5-s + (−1.09 + 2.56i)6-s + (−2.03 − 1.69i)7-s + 2.26·8-s + (−2.07 − 2.16i)9-s + (−0.805 + 1.39i)10-s + (1.08 + 1.87i)11-s + (0.402 − 0.945i)12-s + (−2.21 − 3.83i)13-s + (3.27 + 2.73i)14-s + (−1.04 − 1.38i)15-s − 4.83·16-s + (−0.752 + 1.30i)17-s + ⋯
L(s)  = 1  − 1.13·2-s + (0.391 − 0.920i)3-s + 0.296·4-s + (0.223 − 0.387i)5-s + (−0.445 + 1.04i)6-s + (−0.767 − 0.640i)7-s + 0.800·8-s + (−0.693 − 0.720i)9-s + (−0.254 + 0.441i)10-s + (0.326 + 0.566i)11-s + (0.116 − 0.272i)12-s + (−0.613 − 1.06i)13-s + (0.874 + 0.729i)14-s + (−0.268 − 0.357i)15-s − 1.20·16-s + (−0.182 + 0.316i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.937 + 0.348i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.937 + 0.348i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.937 + 0.348i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ -0.937 + 0.348i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0902073 - 0.501267i\)
\(L(\frac12)\) \(\approx\) \(0.0902073 - 0.501267i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.678 + 1.59i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (2.03 + 1.69i)T \)
good2 \( 1 + 1.61T + 2T^{2} \)
11 \( 1 + (-1.08 - 1.87i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.21 + 3.83i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.752 - 1.30i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.165 - 0.286i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.949 + 1.64i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.99 - 3.45i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 9.75T + 31T^{2} \)
37 \( 1 + (1.09 + 1.90i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.44 + 5.96i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-6.16 + 10.6i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 1.81T + 47T^{2} \)
53 \( 1 + (-1.32 + 2.29i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 7.57T + 59T^{2} \)
61 \( 1 + 4.87T + 61T^{2} \)
67 \( 1 - 13.2T + 67T^{2} \)
71 \( 1 - 14.9T + 71T^{2} \)
73 \( 1 + (5.31 - 9.20i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 5.11T + 79T^{2} \)
83 \( 1 + (-2.99 + 5.19i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (2.99 + 5.18i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3.11 + 5.39i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.94934678372290329985406487997, −10.06260421409877016554140809995, −9.256606209753260429196439336963, −8.504616903740053779492232445456, −7.42055219618603560251490809890, −6.94553244559354415130844540200, −5.43671161817667137074768541689, −3.73626022828928315331154583866, −2.00990996198543770200997922979, −0.49709619075966647822953806374, 2.28554917651463349691949455198, 3.68496373038973317641191073859, 5.03108625827348738147500550705, 6.38830501818687527568030717290, 7.57408409577458086979075366887, 8.710198235684350373020805284916, 9.454874130252589186864791594553, 9.708028011921428538706708977811, 10.91684316771978908052244847726, 11.57647011098751918277003499452

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