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Properties

Label 2-315-63.47-c1-0-29
Degree $2$
Conductor $315$
Sign $0.0232 + 0.999i$
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  + (1.87 − 1.08i)2-s + (0.865 − 1.50i)3-s + (1.33 − 2.31i)4-s + 5-s + (−0.00132 − 3.74i)6-s + (−1.75 + 1.98i)7-s − 1.46i·8-s + (−1.50 − 2.59i)9-s + (1.87 − 1.08i)10-s − 1.72i·11-s + (−2.31 − 4.01i)12-s + (−2.89 + 1.66i)13-s + (−1.13 + 5.60i)14-s + (0.865 − 1.50i)15-s + (1.09 + 1.89i)16-s + (2.47 + 4.28i)17-s + ⋯
L(s)  = 1  + (1.32 − 0.764i)2-s + (0.499 − 0.866i)3-s + (0.669 − 1.15i)4-s + 0.447·5-s + (−0.000539 − 1.52i)6-s + (−0.661 + 0.749i)7-s − 0.517i·8-s + (−0.500 − 0.865i)9-s + (0.592 − 0.341i)10-s − 0.520i·11-s + (−0.669 − 1.15i)12-s + (−0.801 + 0.462i)13-s + (−0.303 + 1.49i)14-s + (0.223 − 0.387i)15-s + (0.273 + 0.474i)16-s + (0.600 + 1.04i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0232 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0232 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.0232 + 0.999i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (236, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.0232 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.04196 - 1.99501i\)
\(L(\frac12)\) \(\approx\) \(2.04196 - 1.99501i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.865 + 1.50i)T \)
5 \( 1 - T \)
7 \( 1 + (1.75 - 1.98i)T \)
good2 \( 1 + (-1.87 + 1.08i)T + (1 - 1.73i)T^{2} \)
11 \( 1 + 1.72iT - 11T^{2} \)
13 \( 1 + (2.89 - 1.66i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.47 - 4.28i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.88 - 1.66i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 2.67iT - 23T^{2} \)
29 \( 1 + (4.72 + 2.72i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (6.73 + 3.89i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.70 - 2.94i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.394 + 0.683i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.82 + 10.0i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.76 - 9.99i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.69 + 3.28i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.45 - 9.45i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.58 - 3.22i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.06 + 1.83i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 1.34iT - 71T^{2} \)
73 \( 1 + (-2.68 + 1.55i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.80 + 11.7i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.95 + 8.58i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (5.36 - 9.28i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.88 - 1.66i)T + (48.5 + 84.0i)T^{2} \)
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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

−12.04137293223608998519234812910, −10.81756893829636124759095979282, −9.628284291579783244141397611417, −8.695222020324225201856184515821, −7.41405640280901615672421608589, −6.05278114401885974270360813623, −5.61548349219494206259428586431, −3.89204945799956195850194679764, −2.85481946820362206472263445994, −1.88969025498365591892400421074, 2.88759503979645015700050579357, 3.78772516596946197563939366211, 4.96587942553776817883758842746, 5.56888718849777381494093419240, 7.07940110729512832494422923616, 7.57636288928971107242922383944, 9.374223100143527936041373499433, 9.814803256164602949606976697976, 10.93364772874919053073492866358, 12.28389579374386532452086359886

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