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Properties

Label 2-315-9.4-c1-0-22
Degree $2$
Conductor $315$
Sign $-0.971 + 0.238i$
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  + (−0.522 − 0.904i)2-s + (0.392 − 1.68i)3-s + (0.454 − 0.787i)4-s + (−0.5 + 0.866i)5-s + (−1.73 + 0.525i)6-s + (0.5 + 0.866i)7-s − 3.03·8-s + (−2.69 − 1.32i)9-s + 1.04·10-s + (−2.85 − 4.94i)11-s + (−1.15 − 1.07i)12-s + (1.04 − 1.81i)13-s + (0.522 − 0.904i)14-s + (1.26 + 1.18i)15-s + (0.676 + 1.17i)16-s + 3.21·17-s + ⋯
L(s)  = 1  + (−0.369 − 0.639i)2-s + (0.226 − 0.973i)3-s + (0.227 − 0.393i)4-s + (−0.223 + 0.387i)5-s + (−0.706 + 0.214i)6-s + (0.188 + 0.327i)7-s − 1.07·8-s + (−0.897 − 0.441i)9-s + 0.330·10-s + (−0.860 − 1.49i)11-s + (−0.332 − 0.310i)12-s + (0.290 − 0.502i)13-s + (0.139 − 0.241i)14-s + (0.326 + 0.305i)15-s + (0.169 + 0.292i)16-s + 0.779·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.971 + 0.238i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.971 + 0.238i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.971 + 0.238i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ -0.971 + 0.238i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.119342 - 0.985775i\)
\(L(\frac12)\) \(\approx\) \(0.119342 - 0.985775i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.392 + 1.68i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
good2 \( 1 + (0.522 + 0.904i)T + (-1 + 1.73i)T^{2} \)
11 \( 1 + (2.85 + 4.94i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.04 + 1.81i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 3.21T + 17T^{2} \)
19 \( 1 + 1.22T + 19T^{2} \)
23 \( 1 + (1.83 - 3.17i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.588 - 1.02i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.29 + 5.70i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 9.74T + 37T^{2} \)
41 \( 1 + (-5.12 + 8.87i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.76 - 8.25i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.46 + 5.99i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 4.52T + 53T^{2} \)
59 \( 1 + (5.91 - 10.2i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.46 + 7.74i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.96 - 5.13i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 14.0T + 71T^{2} \)
73 \( 1 - 5.49T + 73T^{2} \)
79 \( 1 + (6.68 + 11.5i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.74 + 3.02i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 16.2T + 89T^{2} \)
97 \( 1 + (1.04 + 1.81i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.24617133931042633227537079250, −10.56150283548793337971683459517, −9.370075548584898448839504603579, −8.307707463510561327274512795138, −7.63752845208306950567720509998, −6.11869015757645816273983011809, −5.69893617948405878453467748220, −3.29768806550002899434182306883, −2.42827566217118307690145466759, −0.77817113777242596726804053584, 2.60126898871472014201299932480, 4.06144515075988093172036505467, 4.97047794359553488275284220634, 6.33972706496209583760410641211, 7.63161570032433088689723385613, 8.154355976696431090770750863023, 9.240241401372608523432341666089, 10.01547709136029692442313272088, 11.02270121987299346921161662334, 12.08838616498891063970326822871

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