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Properties

Label 2-315-315.4-c1-0-22
Degree $2$
Conductor $315$
Sign $0.483 + 0.875i$
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  + (−0.866 − 0.5i)2-s + 1.73·3-s + (−0.500 − 0.866i)4-s + (1 − 2i)5-s + (−1.49 − 0.866i)6-s + (0.866 + 2.5i)7-s + 3i·8-s + 2.99·9-s + (−1.86 + 1.23i)10-s + 6·11-s + (−0.866 − 1.49i)12-s + (−3.46 − 2i)13-s + (0.500 − 2.59i)14-s + (1.73 − 3.46i)15-s + (0.500 − 0.866i)16-s + (−1.73 − i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + 1.00·3-s + (−0.250 − 0.433i)4-s + (0.447 − 0.894i)5-s + (−0.612 − 0.353i)6-s + (0.327 + 0.944i)7-s + 1.06i·8-s + 0.999·9-s + (−0.590 + 0.389i)10-s + 1.80·11-s + (−0.250 − 0.433i)12-s + (−0.960 − 0.554i)13-s + (0.133 − 0.694i)14-s + (0.447 − 0.894i)15-s + (0.125 − 0.216i)16-s + (−0.420 − 0.242i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.483 + 0.875i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.483 + 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.483 + 0.875i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.483 + 0.875i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.20648 - 0.711947i\)
\(L(\frac12)\) \(\approx\) \(1.20648 - 0.711947i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 1.73T \)
5 \( 1 + (-1 + 2i)T \)
7 \( 1 + (-0.866 - 2.5i)T \)
good2 \( 1 + (0.866 + 0.5i)T + (1 + 1.73i)T^{2} \)
11 \( 1 - 6T + 11T^{2} \)
13 \( 1 + (3.46 + 2i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.73 + i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (3 + 5.19i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 3iT - 23T^{2} \)
29 \( 1 + (1 + 1.73i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2 - 3.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (6.92 - 4i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (1 - 1.73i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.866 - 0.5i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.59 + 1.5i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-10.3 - 6i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2 - 3.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.5 - 6.06i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.06 + 3.5i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 4T + 71T^{2} \)
73 \( 1 + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (7 - 12.1i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.46 - 2i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (1.5 + 2.59i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.73 + i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.55981619489762695797830304446, −10.16724629522007198385099601818, −9.311405639516531243880197818431, −8.958428636738844619955239356839, −8.263940500713275156923381725042, −6.74123410997559027340348339404, −5.33189844135013584119812763553, −4.40442794404849228821731066920, −2.49441936939358577103692945835, −1.40762383048239847299199665577, 1.86224008991233329012157758872, 3.62533051785185703990523860286, 4.23702119251192064689376742855, 6.64617970890918861813327199336, 7.01729054300237932491353457685, 8.050574041233869949281626929797, 8.987824599706140055202899610739, 9.765238087185316169222789417328, 10.47028106353820368393913852220, 11.83966487126700602724876645978

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