Properties

Label 2-315-7.2-c1-0-2
Degree $2$
Conductor $315$
Sign $-0.795 - 0.605i$
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 + 2.36i)2-s + (−2.73 − 4.73i)4-s + (0.5 − 0.866i)5-s + (0.866 + 2.5i)7-s + 9.46·8-s + (1.36 + 2.36i)10-s + (0.366 + 0.633i)11-s + 2.26·13-s + (−7.09 − 1.36i)14-s + (−7.46 + 12.9i)16-s + (1.63 + 2.83i)17-s + (−2.23 + 3.86i)19-s − 5.46·20-s − 2·22-s + (−2.36 + 4.09i)23-s + ⋯
L(s)  = 1  + (−0.965 + 1.67i)2-s + (−1.36 − 2.36i)4-s + (0.223 − 0.387i)5-s + (0.327 + 0.944i)7-s + 3.34·8-s + (0.431 + 0.748i)10-s + (0.110 + 0.191i)11-s + 0.629·13-s + (−1.89 − 0.365i)14-s + (−1.86 + 3.23i)16-s + (0.396 + 0.686i)17-s + (−0.512 + 0.886i)19-s − 1.22·20-s − 0.426·22-s + (−0.493 + 0.854i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.795 - 0.605i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.795 - 0.605i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.795 - 0.605i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (226, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ -0.795 - 0.605i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.246650 + 0.731591i\)
\(L(\frac12)\) \(\approx\) \(0.246650 + 0.731591i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (-0.866 - 2.5i)T \)
good2 \( 1 + (1.36 - 2.36i)T + (-1 - 1.73i)T^{2} \)
11 \( 1 + (-0.366 - 0.633i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 2.26T + 13T^{2} \)
17 \( 1 + (-1.63 - 2.83i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.23 - 3.86i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.36 - 4.09i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 4.19T + 29T^{2} \)
31 \( 1 + (-0.232 - 0.401i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.59 + 2.76i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 0.732T + 41T^{2} \)
43 \( 1 - 3.19T + 43T^{2} \)
47 \( 1 + (-1 + 1.73i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6.19 - 10.7i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.0980 + 0.169i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2 - 3.46i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.33 - 12.6i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 6.19T + 71T^{2} \)
73 \( 1 + (6.33 + 10.9i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.69 + 6.40i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 15.1T + 83T^{2} \)
89 \( 1 + (-7.56 + 13.0i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 14.9T + 97T^{2} \)
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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

−12.03882114000280052014677850754, −10.63850407840027099973351322836, −9.789629820044307308509210263770, −8.839593346214729364914012296219, −8.329964424442525961963791499918, −7.39519618523145091264564336695, −6.01347124749460117831238023397, −5.72282529615036722653565143202, −4.37643004357466078731241563937, −1.56172082471982527027468279590, 0.871949197295696055362585111969, 2.41746114642680862324084948007, 3.61171018310801935757337566004, 4.66133608573315113832298876367, 6.76110145095374915212952486085, 7.88158657293636644475976775724, 8.701856107109919549803919358369, 9.728917208375039060074246844034, 10.47343553936975622848395163503, 11.09702021534657120755990433314

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