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Properties

Label 2-315-63.16-c1-0-26
Degree $2$
Conductor $315$
Sign $-0.663 - 0.748i$
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  + (−0.949 − 1.64i)2-s + (−1.72 + 0.110i)3-s + (−0.804 + 1.39i)4-s + 5-s + (1.82 + 2.73i)6-s + (0.880 − 2.49i)7-s − 0.741·8-s + (2.97 − 0.383i)9-s + (−0.949 − 1.64i)10-s − 4.95·11-s + (1.23 − 2.49i)12-s + (−2.81 − 4.88i)13-s + (−4.94 + 0.921i)14-s + (−1.72 + 0.110i)15-s + (2.31 + 4.00i)16-s + (1.93 + 3.35i)17-s + ⋯
L(s)  = 1  + (−0.671 − 1.16i)2-s + (−0.997 + 0.0640i)3-s + (−0.402 + 0.696i)4-s + 0.447·5-s + (0.744 + 1.11i)6-s + (0.332 − 0.943i)7-s − 0.262·8-s + (0.991 − 0.127i)9-s + (−0.300 − 0.520i)10-s − 1.49·11-s + (0.356 − 0.721i)12-s + (−0.781 − 1.35i)13-s + (−1.32 + 0.246i)14-s + (−0.446 + 0.0286i)15-s + (0.578 + 1.00i)16-s + (0.470 + 0.814i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.663 - 0.748i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.663 - 0.748i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.663 - 0.748i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ -0.663 - 0.748i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.121622 + 0.270208i\)
\(L(\frac12)\) \(\approx\) \(0.121622 + 0.270208i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.72 - 0.110i)T \)
5 \( 1 - T \)
7 \( 1 + (-0.880 + 2.49i)T \)
good2 \( 1 + (0.949 + 1.64i)T + (-1 + 1.73i)T^{2} \)
11 \( 1 + 4.95T + 11T^{2} \)
13 \( 1 + (2.81 + 4.88i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.93 - 3.35i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.540 - 0.935i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 5.37T + 23T^{2} \)
29 \( 1 + (2.83 - 4.90i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.212 - 0.367i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.891 + 1.54i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (4.04 + 7.01i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.60 - 6.23i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (5.30 + 9.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.94 - 3.36i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.14 + 7.18i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.02 - 5.24i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.670 + 1.16i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 5.02T + 71T^{2} \)
73 \( 1 + (5.89 + 10.2i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.12 + 7.14i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.91 - 5.04i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-8.26 + 14.3i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.81 + 8.34i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.70393283534177636849767789076, −10.30812643054931259573904557210, −9.942844191950864072466193608133, −8.279627577873370932718676449398, −7.40942807089268337281907874498, −5.90373038643751273737379141096, −5.04739253160659023332209281334, −3.46293542001891735783646450223, −1.86642128726918645131155878231, −0.28283133867213547464102370890, 2.32947804323236763626012360917, 4.85332003976523645446099806607, 5.56092383706931049237495960181, 6.44070461618065542110447634374, 7.40980072560536231255806554394, 8.242590064355746817056135387265, 9.497343105080194919360821190946, 10.01160068944936403992413396397, 11.46810726352937866862162594456, 12.02315640866940864949277316907

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