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Properties

Label 2-315-63.38-c1-0-30
Degree $2$
Conductor $315$
Sign $-0.233 - 0.972i$
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  − 1.93i·2-s + (−1.06 − 1.36i)3-s − 1.74·4-s + (−0.5 + 0.866i)5-s + (−2.63 + 2.06i)6-s + (−2.29 − 1.31i)7-s − 0.490i·8-s + (−0.718 + 2.91i)9-s + (1.67 + 0.967i)10-s + (−4.11 + 2.37i)11-s + (1.86 + 2.38i)12-s + (1.44 − 0.834i)13-s + (−2.55 + 4.43i)14-s + (1.71 − 0.243i)15-s − 4.44·16-s + (−0.551 + 0.954i)17-s + ⋯
L(s)  = 1  − 1.36i·2-s + (−0.616 − 0.787i)3-s − 0.873·4-s + (−0.223 + 0.387i)5-s + (−1.07 + 0.844i)6-s + (−0.866 − 0.498i)7-s − 0.173i·8-s + (−0.239 + 0.970i)9-s + (0.530 + 0.306i)10-s + (−1.24 + 0.716i)11-s + (0.538 + 0.687i)12-s + (0.400 − 0.231i)13-s + (−0.682 + 1.18i)14-s + (0.442 − 0.0628i)15-s − 1.11·16-s + (−0.133 + 0.231i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.233 - 0.972i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.233 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.233 - 0.972i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ -0.233 - 0.972i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.256852 + 0.325881i\)
\(L(\frac12)\) \(\approx\) \(0.256852 + 0.325881i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.06 + 1.36i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (2.29 + 1.31i)T \)
good2 \( 1 + 1.93iT - 2T^{2} \)
11 \( 1 + (4.11 - 2.37i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.44 + 0.834i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.551 - 0.954i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.85 + 2.22i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (7.43 + 4.28i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.66 - 1.54i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 3.96iT - 31T^{2} \)
37 \( 1 + (-0.755 - 1.30i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.80 + 6.58i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.0488 + 0.0845i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 5.51T + 47T^{2} \)
53 \( 1 + (11.1 + 6.42i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + 13.3T + 59T^{2} \)
61 \( 1 + 10.7iT - 61T^{2} \)
67 \( 1 + 5.06T + 67T^{2} \)
71 \( 1 + 4.66iT - 71T^{2} \)
73 \( 1 + (-3.66 - 2.11i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + 6.21T + 79T^{2} \)
83 \( 1 + (-2.60 + 4.50i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-7.15 - 12.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.33 - 0.768i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.90265251053279144225783423253, −10.46106493545810786577492730037, −9.642394303756325822186102692787, −8.025920460151909067938174747406, −7.09632828555732526516507577595, −6.13661049873802085996891397525, −4.62931111630554799258591546488, −3.22730445894020981470359010510, −2.12647246184461693091107586741, −0.30350076559158630558826785032, 3.22002187291018107234004315061, 4.68844418061289057812332315413, 5.74980127186731137093726179290, 6.10119064525964557178694194782, 7.50158400625737560125968707816, 8.448638657542732905181496074981, 9.345620339890689988229365913726, 10.28912489028999247425998779567, 11.43189954334667611849355623388, 12.23185896640096692298751693606

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