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Properties

Label 2-315-63.38-c1-0-13
Degree $2$
Conductor $315$
Sign $-0.202 - 0.979i$
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  + 1.71i·2-s + (1.72 + 0.189i)3-s − 0.926·4-s + (−0.5 + 0.866i)5-s + (−0.324 + 2.94i)6-s + (1.44 − 2.21i)7-s + 1.83i·8-s + (2.92 + 0.653i)9-s + (−1.48 − 0.855i)10-s + (−0.497 + 0.287i)11-s + (−1.59 − 0.175i)12-s + (−0.130 + 0.0752i)13-s + (3.79 + 2.46i)14-s + (−1.02 + 1.39i)15-s − 4.99·16-s + (−0.586 + 1.01i)17-s + ⋯
L(s)  = 1  + 1.20i·2-s + (0.993 + 0.109i)3-s − 0.463·4-s + (−0.223 + 0.387i)5-s + (−0.132 + 1.20i)6-s + (0.544 − 0.838i)7-s + 0.649i·8-s + (0.975 + 0.217i)9-s + (−0.468 − 0.270i)10-s + (−0.149 + 0.0865i)11-s + (−0.460 − 0.0508i)12-s + (−0.0361 + 0.0208i)13-s + (1.01 + 0.658i)14-s + (−0.264 + 0.360i)15-s − 1.24·16-s + (−0.142 + 0.246i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.202 - 0.979i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.202 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.202 - 0.979i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ -0.202 - 0.979i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.18411 + 1.45382i\)
\(L(\frac12)\) \(\approx\) \(1.18411 + 1.45382i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.72 - 0.189i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (-1.44 + 2.21i)T \)
good2 \( 1 - 1.71iT - 2T^{2} \)
11 \( 1 + (0.497 - 0.287i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.130 - 0.0752i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.586 - 1.01i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.00148 + 0.000858i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (6.46 + 3.73i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-5.81 - 3.35i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 6.63iT - 31T^{2} \)
37 \( 1 + (0.718 + 1.24i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (5.37 + 9.30i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.00 - 5.20i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 8.44T + 47T^{2} \)
53 \( 1 + (-9.60 - 5.54i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 - 10.2T + 59T^{2} \)
61 \( 1 + 7.35iT - 61T^{2} \)
67 \( 1 + 8.20T + 67T^{2} \)
71 \( 1 + 0.0708iT - 71T^{2} \)
73 \( 1 + (-10.2 - 5.91i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 - 3.31T + 79T^{2} \)
83 \( 1 + (-4.47 + 7.75i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-0.699 - 1.21i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-14.9 - 8.61i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.93958766761427495651133838448, −10.76718266668727978767012744679, −9.985162145374605087837559032046, −8.601392704109532120902221748871, −8.010100635370197846134615728093, −7.24622373052637955635618225374, −6.40820998114194850586238330167, −4.89065836071319100004994415091, −3.85637639038364989188230486004, −2.22363283592095736276994891460, 1.58092611490999642861489704444, 2.64773086200353192744851771418, 3.76378573435219360939656388294, 4.96755771146581602585188293127, 6.63785514328283883351921631612, 7.971896486564465788833706972102, 8.676459278868282390134191298843, 9.657877776469424143744022596927, 10.38401465517796876522665706050, 11.77598854813519711610425212277

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