↑

Properties

Label 2-315-35.4-c1-0-13
Degree $2$
Conductor $315$
Sign $-0.414 + 0.910i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.515 + 0.297i)2-s + (−0.822 + 1.42i)4-s + (−2.19 − 0.437i)5-s + (−1.68 + 2.03i)7-s − 2.16i·8-s + (1.26 − 0.426i)10-s + (2.68 − 4.65i)11-s − 4.07i·13-s + (0.261 − 1.55i)14-s + (−0.999 − 1.73i)16-s + (−4.27 − 2.46i)17-s + (−0.322 − 0.559i)19-s + (2.42 − 2.76i)20-s + 3.20i·22-s + (−5.30 + 3.06i)23-s + ⋯
L(s)  = 1  + (−0.364 + 0.210i)2-s + (−0.411 + 0.712i)4-s + (−0.980 − 0.195i)5-s + (−0.636 + 0.771i)7-s − 0.767i·8-s + (0.398 − 0.135i)10-s + (0.810 − 1.40i)11-s − 1.13i·13-s + (0.0698 − 0.415i)14-s + (−0.249 − 0.433i)16-s + (−1.03 − 0.598i)17-s + (−0.0740 − 0.128i)19-s + (0.542 − 0.618i)20-s + 0.682i·22-s + (−1.10 + 0.638i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.414 + 0.910i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.414 + 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.414 + 0.910i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ -0.414 + 0.910i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.130281 - 0.202444i\)
\(L(\frac12)\) \(\approx\) \(0.130281 - 0.202444i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (2.19 + 0.437i)T \)
7 \( 1 + (1.68 - 2.03i)T \)
good2 \( 1 + (0.515 - 0.297i)T + (1 - 1.73i)T^{2} \)
11 \( 1 + (-2.68 + 4.65i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 4.07iT - 13T^{2} \)
17 \( 1 + (4.27 + 2.46i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.322 + 0.559i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (5.30 - 3.06i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 2.42T + 29T^{2} \)
31 \( 1 + (2.5 - 4.33i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.760 - 0.439i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 11.2T + 41T^{2} \)
43 \( 1 + 10.7iT - 43T^{2} \)
47 \( 1 + (-1.87 + 1.08i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6.85 - 3.95i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.16 - 7.21i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.177 + 0.306i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.01 + 1.16i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 2.42T + 71T^{2} \)
73 \( 1 + (7.82 + 4.51i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.96 - 5.14i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 11.6iT - 83T^{2} \)
89 \( 1 + (2.68 + 4.65i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 6.71iT - 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.66197948986027950289016691206, −10.39038601965258828958300542264, −8.919902525036526464979400234672, −8.780039089973224302486309075598, −7.70340249103593171768142713945, −6.66499916153940463505557334354, −5.40176205594638905670859520086, −3.86920096459963239156885127639, −3.13182400362355767395313666100, −0.19463935501556121823008584717, 1.85857259400769865468098174735, 4.02527555371641886626570301488, 4.49295581793166366029991799559, 6.38304730976467103441369022438, 7.04668241480252083446034507305, 8.295882136858480094105490615701, 9.347246033982738931031465029266, 10.01431717928508183578166286802, 10.96249541968700261262110773168, 11.77366006759841810600098438684

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