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Properties

Label 2-315-63.47-c1-0-21
Degree $2$
Conductor $315$
Sign $-0.576 + 0.816i$
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  + (0.343 − 0.198i)2-s + (−1.65 + 0.520i)3-s + (−0.921 + 1.59i)4-s − 5-s + (−0.464 + 0.506i)6-s + (−0.324 − 2.62i)7-s + 1.52i·8-s + (2.45 − 1.71i)9-s + (−0.343 + 0.198i)10-s − 2.24i·11-s + (0.691 − 3.11i)12-s + (−3.23 + 1.86i)13-s + (−0.632 − 0.838i)14-s + (1.65 − 0.520i)15-s + (−1.53 − 2.66i)16-s + (−3.44 − 5.96i)17-s + ⋯
L(s)  = 1  + (0.243 − 0.140i)2-s + (−0.953 + 0.300i)3-s + (−0.460 + 0.797i)4-s − 0.447·5-s + (−0.189 + 0.206i)6-s + (−0.122 − 0.992i)7-s + 0.539i·8-s + (0.819 − 0.573i)9-s + (−0.108 + 0.0627i)10-s − 0.676i·11-s + (0.199 − 0.899i)12-s + (−0.896 + 0.517i)13-s + (−0.169 − 0.224i)14-s + (0.426 − 0.134i)15-s + (−0.384 − 0.666i)16-s + (−0.835 − 1.44i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.576 + 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.576 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.576 + 0.816i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (236, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ -0.576 + 0.816i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.145171 - 0.280289i\)
\(L(\frac12)\) \(\approx\) \(0.145171 - 0.280289i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.65 - 0.520i)T \)
5 \( 1 + T \)
7 \( 1 + (0.324 + 2.62i)T \)
good2 \( 1 + (-0.343 + 0.198i)T + (1 - 1.73i)T^{2} \)
11 \( 1 + 2.24iT - 11T^{2} \)
13 \( 1 + (3.23 - 1.86i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (3.44 + 5.96i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.97 + 1.14i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 5.41iT - 23T^{2} \)
29 \( 1 + (-1.44 - 0.837i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.61 - 1.50i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.18 - 5.52i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.57 - 2.72i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.90 - 6.76i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.33 + 5.76i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.20 + 1.84i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.61 - 2.80i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (9.75 - 5.63i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.08 + 10.5i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 3.37iT - 71T^{2} \)
73 \( 1 + (10.3 - 5.98i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.62 + 4.54i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-8.61 + 14.9i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-2.58 + 4.46i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (4.94 + 2.85i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.51237926379505809474035022154, −10.62143243759979555556711418804, −9.574042593995496430628413514132, −8.507903055244816952339813423911, −7.28406901616700463405414944989, −6.59483149852404449855421979949, −4.78480931993607706650110607574, −4.45114317903787692795621869638, −3.07515481799657923817663018388, −0.23266779123163028134932449672, 1.91926022645411670030752663088, 4.15119630794669189784250380355, 5.16616056087051875970434938780, 5.94314403858544733060148373375, 6.88493198848348648134762429006, 8.113129202122669618159366430960, 9.336652751352178518101289298812, 10.24396971916954519953404363594, 11.03825477147961429117965874085, 12.22507317439455354868042408018

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