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Properties

Label 2-315-315.103-c1-0-10
Degree $2$
Conductor $315$
Sign $-0.711 - 0.702i$
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  + (0.366 − 0.366i)2-s + (−0.866 + 1.5i)3-s + 1.73i·4-s + (0.133 + 2.23i)5-s + (0.232 + 0.866i)6-s + (−1.73 − 2i)7-s + (1.36 + 1.36i)8-s + (−1.5 − 2.59i)9-s + (0.866 + 0.767i)10-s + (−0.732 + 1.26i)11-s + (−2.59 − 1.49i)12-s + (0.267 + i)13-s + (−1.36 − 0.0980i)14-s + (−3.46 − 1.73i)15-s − 2.46·16-s + (0.732 − 2.73i)17-s + ⋯
L(s)  = 1  + (0.258 − 0.258i)2-s + (−0.499 + 0.866i)3-s + 0.866i·4-s + (0.0599 + 0.998i)5-s + (0.0947 + 0.353i)6-s + (−0.654 − 0.755i)7-s + (0.482 + 0.482i)8-s + (−0.5 − 0.866i)9-s + (0.273 + 0.242i)10-s + (−0.220 + 0.382i)11-s + (−0.749 − 0.433i)12-s + (0.0743 + 0.277i)13-s + (−0.365 − 0.0262i)14-s + (−0.894 − 0.447i)15-s − 0.616·16-s + (0.177 − 0.662i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.711 - 0.702i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.711 - 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.711 - 0.702i$
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: \((2,\ 315,\ (\ :1/2),\ -0.711 - 0.702i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.377464 + 0.919179i\)
\(L(\frac12)\) \(\approx\) \(0.377464 + 0.919179i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.133 - 2.23i)T \)
7 \( 1 + (1.73 + 2i)T \)
good2 \( 1 + (-0.366 + 0.366i)T - 2iT^{2} \)
11 \( 1 + (0.732 - 1.26i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.267 - i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + (-0.732 + 2.73i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (3.36 - 5.83i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.598 + 2.23i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (-6 + 3.46i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 10.1iT - 31T^{2} \)
37 \( 1 + (-1.09 - 4.09i)T + (-32.0 + 18.5i)T^{2} \)
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 \( 1 + (-6.29 - 6.29i)T + 47iT^{2} \)
53 \( 1 + (-0.901 + 3.36i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + 3.46T + 59T^{2} \)
61 \( 1 + 9.39iT - 61T^{2} \)
67 \( 1 + (-6.09 + 6.09i)T - 67iT^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + (0.633 + 0.169i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 - 11.4iT - 79T^{2} \)
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

−12.00482441387792214291968890800, −10.91726512439979853709450995087, −10.39697470254033133779459746189, −9.523068271236722943021754200657, −8.176952247421851009115722207049, −7.04025135417937347343022941242, −6.23254840174822251725076940278, −4.63802104591360219037439283772, −3.74081505988977060870912655182, −2.80814351750477184797216788359, 0.69786328041883202833186538127, 2.31968513302170095064347682621, 4.51476901523373717658871278548, 5.69419573641815422222293760953, 6.01779452003927507207944251694, 7.24617993888382265420238527833, 8.514923565334094930816842165409, 9.304561444333467435764085121330, 10.49672543339316159098724642669, 11.41689776116056456814722925950

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