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Properties

Label 2-315-315.103-c1-0-2
Degree $2$
Conductor $315$
Sign $-0.366 + 0.930i$
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  + (−1.36 + 1.36i)2-s + (0.866 + 1.5i)3-s − 1.73i·4-s + (−1.23 + 1.86i)5-s + (−3.23 − 0.866i)6-s + (−2.5 − 0.866i)7-s + (−0.366 − 0.366i)8-s + (−1.5 + 2.59i)9-s + (−0.866 − 4.23i)10-s + (1 − 1.73i)11-s + (2.59 − 1.49i)12-s + (−1.36 − 5.09i)13-s + (4.59 − 2.23i)14-s + (−3.86 − 0.232i)15-s + 4.46·16-s + (−2 + 7.46i)17-s + ⋯
L(s)  = 1  + (−0.965 + 0.965i)2-s + (0.499 + 0.866i)3-s − 0.866i·4-s + (−0.550 + 0.834i)5-s + (−1.31 − 0.353i)6-s + (−0.944 − 0.327i)7-s + (−0.129 − 0.129i)8-s + (−0.5 + 0.866i)9-s + (−0.273 − 1.33i)10-s + (0.301 − 0.522i)11-s + (0.749 − 0.433i)12-s + (−0.378 − 1.41i)13-s + (1.22 − 0.596i)14-s + (−0.998 − 0.0599i)15-s + 1.11·16-s + (−0.485 + 1.81i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.366 + 0.930i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.366 + 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.366 + 0.930i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ -0.366 + 0.930i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.193877 - 0.284861i\)
\(L(\frac12)\) \(\approx\) \(0.193877 - 0.284861i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.866 - 1.5i)T \)
5 \( 1 + (1.23 - 1.86i)T \)
7 \( 1 + (2.5 + 0.866i)T \)
good2 \( 1 + (1.36 - 1.36i)T - 2iT^{2} \)
11 \( 1 + (-1 + 1.73i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.36 + 5.09i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + (2 - 7.46i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (1.36 - 2.36i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.901 - 3.36i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (-3.40 + 1.96i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 1.46iT - 31T^{2} \)
37 \( 1 + (-0.169 - 0.633i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (2.59 + 1.5i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.866 - 3.23i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (5.36 + 5.36i)T + 47iT^{2} \)
53 \( 1 + (1 - 3.73i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + 15.1T + 59T^{2} \)
61 \( 1 - 2.92iT - 61T^{2} \)
67 \( 1 + (4.46 - 4.46i)T - 67iT^{2} \)
71 \( 1 - 4.73T + 71T^{2} \)
73 \( 1 + (-3.46 - 0.928i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 - 4.53iT - 79T^{2} \)
83 \( 1 + (-11.3 - 3.03i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 + (7.46 - 12.9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.90 - 14.5i)T + (-84.0 - 48.5i)T^{2} \)
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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

−12.26443131283426876013514281997, −10.68808389917286706184424531762, −10.33500055252488346251759721202, −9.443397206277839595066956679127, −8.280643932504788946015645474593, −7.889226659468453056580134016100, −6.62970116875380130347922743088, −5.82199345093877289200299098083, −3.87473427407453797701480896846, −3.12828641498508244122947207559, 0.30592219631651502040335996303, 1.93735988007472274810727168499, 3.05476907981961846827518004794, 4.66290640874028839870997826668, 6.47073801902588584579930887830, 7.36008199460723462325866289347, 8.585852970804462460954165968306, 9.240359554657303233579928659480, 9.625935290962573434583415126633, 11.25474036652175201936626481452

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