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Properties

Label 2-315-63.5-c1-0-19
Degree $2$
Conductor $315$
Sign $0.999 + 0.0138i$
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  + 0.929i·2-s + (0.479 − 1.66i)3-s + 1.13·4-s + (0.5 + 0.866i)5-s + (1.54 + 0.445i)6-s + (0.0655 − 2.64i)7-s + 2.91i·8-s + (−2.54 − 1.59i)9-s + (−0.804 + 0.464i)10-s + (4.54 + 2.62i)11-s + (0.544 − 1.89i)12-s + (−0.516 − 0.298i)13-s + (2.45 + 0.0609i)14-s + (1.68 − 0.417i)15-s − 0.435·16-s + (−1.80 − 3.12i)17-s + ⋯
L(s)  = 1  + 0.657i·2-s + (0.276 − 0.960i)3-s + 0.568·4-s + (0.223 + 0.387i)5-s + (0.631 + 0.181i)6-s + (0.0247 − 0.999i)7-s + 1.03i·8-s + (−0.847 − 0.531i)9-s + (−0.254 + 0.146i)10-s + (1.36 + 0.790i)11-s + (0.157 − 0.546i)12-s + (−0.143 − 0.0826i)13-s + (0.656 + 0.0162i)14-s + (0.434 − 0.107i)15-s − 0.108·16-s + (−0.437 − 0.757i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0138i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0138i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.999 + 0.0138i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.999 + 0.0138i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.74030 - 0.0120109i\)
\(L(\frac12)\) \(\approx\) \(1.74030 - 0.0120109i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.479 + 1.66i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (-0.0655 + 2.64i)T \)
good2 \( 1 - 0.929iT - 2T^{2} \)
11 \( 1 + (-4.54 - 2.62i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.516 + 0.298i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.80 + 3.12i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.80 + 1.62i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.38 + 1.37i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.53 + 2.04i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 6.39iT - 31T^{2} \)
37 \( 1 + (-0.988 + 1.71i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (4.89 - 8.47i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.604 - 1.04i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 5.73T + 47T^{2} \)
53 \( 1 + (8.79 - 5.08i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 1.32T + 59T^{2} \)
61 \( 1 + 10.0iT - 61T^{2} \)
67 \( 1 + 9.26T + 67T^{2} \)
71 \( 1 - 11.6iT - 71T^{2} \)
73 \( 1 + (-12.2 + 7.05i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + 12.9T + 79T^{2} \)
83 \( 1 + (5.40 + 9.36i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (4.26 - 7.39i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-11.2 + 6.49i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.65172938922800786187540389983, −10.93732819445083451064185901021, −9.679997182698464767457693615079, −8.537590598508796720105735942849, −7.47755559533721362279759206184, −6.75296213634783974740627820428, −6.42228974691889317146278950835, −4.69451383771094740373159969679, −2.99598593563819940924737618081, −1.59720906723468838424837719733, 1.89039017157693431153480925511, 3.20297360533732566249971907190, 4.23986573707253361834399971090, 5.69709313456986041886315253785, 6.53286801641715303063589654957, 8.318867367616001587654137809836, 9.024001685477499295927177272709, 9.812377868759584439111758110518, 10.83653334388942665755774738156, 11.55594093847626375097064802935

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