Properties

Label 2-315-315.283-c1-0-12
Degree $2$
Conductor $315$
Sign $0.787 - 0.616i$
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  + (0.133 + 0.5i)2-s − 1.73·3-s + (1.5 − 0.866i)4-s + (2 + i)5-s + (−0.232 − 0.866i)6-s + (−2.5 + 0.866i)7-s + (1.36 + 1.36i)8-s + 2.99·9-s + (−0.232 + 1.13i)10-s + 1.46·11-s + (−2.59 + 1.49i)12-s + (−0.267 − i)13-s + (−0.767 − 1.13i)14-s + (−3.46 − 1.73i)15-s + (1.23 − 2.13i)16-s + (2.73 − 0.732i)17-s + ⋯
L(s)  = 1  + (0.0947 + 0.353i)2-s − 1.00·3-s + (0.750 − 0.433i)4-s + (0.894 + 0.447i)5-s + (−0.0947 − 0.353i)6-s + (−0.944 + 0.327i)7-s + (0.482 + 0.482i)8-s + 0.999·9-s + (−0.0733 + 0.358i)10-s + 0.441·11-s + (−0.749 + 0.433i)12-s + (−0.0743 − 0.277i)13-s + (−0.205 − 0.303i)14-s + (−0.894 − 0.447i)15-s + (0.308 − 0.533i)16-s + (0.662 − 0.177i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.787 - 0.616i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.787 - 0.616i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.787 - 0.616i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (283, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.787 - 0.616i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.28727 + 0.444330i\)
\(L(\frac12)\) \(\approx\) \(1.28727 + 0.444330i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2 - i)T \)
7 \( 1 + (2.5 - 0.866i)T \)
good2 \( 1 + (-0.133 - 0.5i)T + (-1.73 + i)T^{2} \)
11 \( 1 - 1.46T + 11T^{2} \)
13 \( 1 + (0.267 + i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + (-2.73 + 0.732i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-3.36 - 5.83i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.63 - 1.63i)T + 23iT^{2} \)
29 \( 1 + (-6 + 3.46i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (8.83 - 5.09i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.09 + 1.09i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (8.19 + 4.73i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.5 + 5.59i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (-8.59 + 2.30i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (3.36 - 0.901i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (1.73 + 3i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (8.13 + 4.69i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (8.33 + 2.23i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + (0.169 + 0.633i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (9.92 + 5.73i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.36 + 0.366i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 + (0.598 + 1.03i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.830 + 3.09i)T + (-84.0 - 48.5i)T^{2} \)
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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

−11.88463705644405079240898094435, −10.55175675541670438460041123634, −10.23292787992998466212253863988, −9.257311323211612110919908467729, −7.41586425351249769546076782423, −6.70259610839748692798263668379, −5.81640516647960740647113602964, −5.36587030356323068513949997206, −3.30990931373935617882119816718, −1.63071287078507208076661067811, 1.30546397913012155492974109565, 3.00536102888674295639694855635, 4.45915260506102175926438040120, 5.74192982056113064527463401039, 6.64661266107380402057013430666, 7.30778070142123944497219917324, 9.052374576806589374346117886438, 9.941102689190855926543496268761, 10.68319457694466879736186173687, 11.62240056203537346094567777684

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