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Properties

Label 2-315-63.25-c1-0-9
Degree $2$
Conductor $315$
Sign $0.958 + 0.284i$
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  − 2.35·2-s + (−0.431 − 1.67i)3-s + 3.56·4-s + (−0.5 + 0.866i)5-s + (1.01 + 3.95i)6-s + (2.33 − 1.25i)7-s − 3.69·8-s + (−2.62 + 1.44i)9-s + (1.17 − 2.04i)10-s + (2.53 + 4.38i)11-s + (−1.53 − 5.98i)12-s + (0.354 + 0.613i)13-s + (−5.49 + 2.95i)14-s + (1.66 + 0.465i)15-s + 1.58·16-s + (−1.67 + 2.90i)17-s + ⋯
L(s)  = 1  − 1.66·2-s + (−0.249 − 0.968i)3-s + 1.78·4-s + (−0.223 + 0.387i)5-s + (0.415 + 1.61i)6-s + (0.880 − 0.473i)7-s − 1.30·8-s + (−0.875 + 0.482i)9-s + (0.373 − 0.646i)10-s + (0.763 + 1.32i)11-s + (−0.444 − 1.72i)12-s + (0.0981 + 0.170i)13-s + (−1.46 + 0.790i)14-s + (0.430 + 0.120i)15-s + 0.396·16-s + (−0.407 + 0.705i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.958 + 0.284i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.958 + 0.284i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.958 + 0.284i$
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: \((2,\ 315,\ (\ :1/2),\ 0.958 + 0.284i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.584081 - 0.0847410i\)
\(L(\frac12)\) \(\approx\) \(0.584081 - 0.0847410i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.431 + 1.67i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (-2.33 + 1.25i)T \)
good2 \( 1 + 2.35T + 2T^{2} \)
11 \( 1 + (-2.53 - 4.38i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.354 - 0.613i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.67 - 2.90i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.99 - 3.45i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.79 + 6.58i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.54 + 6.13i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 5.32T + 31T^{2} \)
37 \( 1 + (-3.70 - 6.41i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.911 - 1.57i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.09 + 1.89i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 10.5T + 47T^{2} \)
53 \( 1 + (-4.88 + 8.45i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 10.0T + 59T^{2} \)
61 \( 1 + 0.146T + 61T^{2} \)
67 \( 1 + 6.82T + 67T^{2} \)
71 \( 1 + 4.39T + 71T^{2} \)
73 \( 1 + (1.10 - 1.92i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 0.556T + 79T^{2} \)
83 \( 1 + (4.40 - 7.62i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-4.50 - 7.80i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.62 - 2.81i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.53014050766479723452263003502, −10.55994063907137776663366236151, −9.818579521535556927301711571766, −8.452907725830860785026683825434, −7.998613692843080350936734481544, −6.98139722152938738669541430320, −6.47044652460128159179264916560, −4.52160780468592127176284757561, −2.27553101518125930810248352436, −1.18160478050788969497620865837, 0.982527046232763007992550004754, 3.03113056084089110072104026408, 4.71763528946240736472095533409, 5.85096143611453994025363210519, 7.21846845725105132575756095383, 8.436362913621296720285250096080, 8.914377917201880350973917654134, 9.543370710230338197014302032193, 10.79044057575951711031045664220, 11.37494043421817007362788112937

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