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Properties

Label 2-315-63.4-c1-0-21
Degree $2$
Conductor $315$
Sign $0.784 - 0.619i$
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  + (−0.415 + 0.719i)2-s + (1.70 − 0.281i)3-s + (0.654 + 1.13i)4-s + 5-s + (−0.507 + 1.34i)6-s + (0.762 − 2.53i)7-s − 2.75·8-s + (2.84 − 0.961i)9-s + (−0.415 + 0.719i)10-s + 2.12·11-s + (1.43 + 1.75i)12-s + (0.552 − 0.956i)13-s + (1.50 + 1.60i)14-s + (1.70 − 0.281i)15-s + (−0.165 + 0.287i)16-s + (−3.19 + 5.53i)17-s + ⋯
L(s)  = 1  + (−0.293 + 0.509i)2-s + (0.986 − 0.162i)3-s + (0.327 + 0.566i)4-s + 0.447·5-s + (−0.207 + 0.549i)6-s + (0.288 − 0.957i)7-s − 0.972·8-s + (0.947 − 0.320i)9-s + (−0.131 + 0.227i)10-s + 0.639·11-s + (0.414 + 0.506i)12-s + (0.153 − 0.265i)13-s + (0.402 + 0.428i)14-s + (0.441 − 0.0726i)15-s + (−0.0414 + 0.0718i)16-s + (−0.775 + 1.34i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.784 - 0.619i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.784 - 0.619i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.784 - 0.619i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (256, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.784 - 0.619i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.67865 + 0.583093i\)
\(L(\frac12)\) \(\approx\) \(1.67865 + 0.583093i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.70 + 0.281i)T \)
5 \( 1 - T \)
7 \( 1 + (-0.762 + 2.53i)T \)
good2 \( 1 + (0.415 - 0.719i)T + (-1 - 1.73i)T^{2} \)
11 \( 1 - 2.12T + 11T^{2} \)
13 \( 1 + (-0.552 + 0.956i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (3.19 - 5.53i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.76 + 4.79i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 7.65T + 23T^{2} \)
29 \( 1 + (0.160 + 0.278i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-5.40 - 9.36i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.76 - 3.05i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (5.63 - 9.75i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.52 + 7.83i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.17 - 2.04i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.02 + 8.71i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.75 + 4.77i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.98 + 5.17i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.120 + 0.208i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 3.28T + 71T^{2} \)
73 \( 1 + (3.39 - 5.87i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.59 + 4.48i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.94 + 5.10i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-2.74 - 4.75i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (8.61 + 14.9i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.83899901745944009210417141607, −10.64081709947779624456657538711, −9.739907395617887798146732243674, −8.448876559860361488425717496374, −8.267533264934961404243503187555, −6.87441126664678715815580680441, −6.47140941022395648835967023941, −4.40602298301236099557022189397, −3.36939509178020025523727216753, −1.87817315999109289655257266296, 1.83539568846211428558807111600, 2.59117967271562363325194737812, 4.20298446176932890436815077677, 5.68574342445655428730976577412, 6.63030432764599093228211662767, 8.071580380637637005195227209370, 9.044096693007973982003558734846, 9.587287906321646817501725212483, 10.41359030716796905092945478795, 11.60760531971097006527735388667

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