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Properties

Label 2-315-9.4-c1-0-3
Degree $2$
Conductor $315$
Sign $0.717 + 0.696i$
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  + (−0.875 − 1.51i)2-s + (−1.16 + 1.28i)3-s + (−0.532 + 0.922i)4-s + (−0.5 + 0.866i)5-s + (2.96 + 0.635i)6-s + (−0.5 − 0.866i)7-s − 1.63·8-s + (−0.304 − 2.98i)9-s + 1.75·10-s + (3.05 + 5.29i)11-s + (−0.567 − 1.75i)12-s + (2.90 − 5.03i)13-s + (−0.875 + 1.51i)14-s + (−0.532 − 1.64i)15-s + (2.49 + 4.32i)16-s + 4.73·17-s + ⋯
L(s)  = 1  + (−0.619 − 1.07i)2-s + (−0.670 + 0.742i)3-s + (−0.266 + 0.461i)4-s + (−0.223 + 0.387i)5-s + (1.21 + 0.259i)6-s + (−0.188 − 0.327i)7-s − 0.578·8-s + (−0.101 − 0.994i)9-s + 0.553·10-s + (0.921 + 1.59i)11-s + (−0.163 − 0.506i)12-s + (0.806 − 1.39i)13-s + (−0.233 + 0.405i)14-s + (−0.137 − 0.425i)15-s + (0.624 + 1.08i)16-s + 1.14·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.717 + 0.696i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.717 + 0.696i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.717 + 0.696i$
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: \((2,\ 315,\ (\ :1/2),\ 0.717 + 0.696i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.712532 - 0.289181i\)
\(L(\frac12)\) \(\approx\) \(0.712532 - 0.289181i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.16 - 1.28i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (0.875 + 1.51i)T + (-1 + 1.73i)T^{2} \)
11 \( 1 + (-3.05 - 5.29i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.90 + 5.03i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 4.73T + 17T^{2} \)
19 \( 1 - 3.68T + 19T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.839 - 1.45i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.47 + 6.02i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 4.49T + 37T^{2} \)
41 \( 1 + (-1.98 + 3.44i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.66 - 9.80i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.08 + 3.61i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 8.30T + 53T^{2} \)
59 \( 1 + (0.418 - 0.724i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.62 - 8.01i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.25 - 3.91i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 10.3T + 71T^{2} \)
73 \( 1 - 12.2T + 73T^{2} \)
79 \( 1 + (4.41 + 7.64i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.17 - 5.49i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 9.66T + 89T^{2} \)
97 \( 1 + (-4.08 - 7.07i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.37181511455271409076496301505, −10.51207054831290464499042944001, −9.922950621814965440332024938783, −9.332370506013815219633733111647, −7.902328017580116541580695960935, −6.61082032603150887426516934354, −5.51683512688016421115284723642, −4.03333851892460593903124765424, −3.07936918503897954517813103856, −1.06647657593498702479766079628, 1.08169215608582161733860673364, 3.44809979515117095330961747289, 5.33371825059881934008330162740, 6.19574183587548839983420449005, 6.81113667408921478389442418238, 7.967563111052044784650511048921, 8.668121926845062824334446127497, 9.478510730525956110366382776536, 11.09985794270793196222315099033, 11.81105370168673426315926406540

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