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Properties

Label 2-315-7.6-c2-0-25
Degree $2$
Conductor $315$
Sign $0.936 + 0.349i$
Analytic cond. $8.58312$
Root an. cond. $2.92969$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.80·2-s + 10.4·4-s − 2.23i·5-s + (2.44 − 6.55i)7-s + 24.6·8-s − 8.50i·10-s − 14.4·11-s + 16.9i·13-s + (9.30 − 24.9i)14-s + 51.8·16-s − 13.0i·17-s + 18.6i·19-s − 23.4i·20-s − 54.9·22-s − 10.3·23-s + ⋯
L(s)  = 1  + 1.90·2-s + 2.61·4-s − 0.447i·5-s + (0.349 − 0.936i)7-s + 3.07·8-s − 0.850i·10-s − 1.31·11-s + 1.30i·13-s + (0.664 − 1.78i)14-s + 3.23·16-s − 0.765i·17-s + 0.983i·19-s − 1.17i·20-s − 2.49·22-s − 0.451·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.936 + 0.349i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.936 + 0.349i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.936 + 0.349i$
Analytic conductor: \(8.58312\)
Root analytic conductor: \(2.92969\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1),\ 0.936 + 0.349i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(4.92655 - 0.888830i\)
\(L(\frac12)\) \(\approx\) \(4.92655 - 0.888830i\)
\(L(2)\) 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 \( 1 + 2.23iT \)
7 \( 1 + (-2.44 + 6.55i)T \)
good2 \( 1 - 3.80T + 4T^{2} \)
11 \( 1 + 14.4T + 121T^{2} \)
13 \( 1 - 16.9iT - 169T^{2} \)
17 \( 1 + 13.0iT - 289T^{2} \)
19 \( 1 - 18.6iT - 361T^{2} \)
23 \( 1 + 10.3T + 529T^{2} \)
29 \( 1 - 13.7T + 841T^{2} \)
31 \( 1 - 42.4iT - 961T^{2} \)
37 \( 1 - 28.7T + 1.36e3T^{2} \)
41 \( 1 - 28.8iT - 1.68e3T^{2} \)
43 \( 1 + 5.84T + 1.84e3T^{2} \)
47 \( 1 + 10.5iT - 2.20e3T^{2} \)
53 \( 1 + 81.9T + 2.80e3T^{2} \)
59 \( 1 + 35.1iT - 3.48e3T^{2} \)
61 \( 1 + 68.4iT - 3.72e3T^{2} \)
67 \( 1 - 47.4T + 4.48e3T^{2} \)
71 \( 1 + 47.6T + 5.04e3T^{2} \)
73 \( 1 + 125. iT - 5.32e3T^{2} \)
79 \( 1 + 129.T + 6.24e3T^{2} \)
83 \( 1 - 42.6iT - 6.88e3T^{2} \)
89 \( 1 + 25.5iT - 7.92e3T^{2} \)
97 \( 1 - 28.7iT - 9.40e3T^{2} \)
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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

−11.64870539993185020290494700039, −10.87047787524665245827471882085, −9.916645769747492590753020918676, −8.093131969244903663230447394032, −7.22675121800998700903896727305, −6.23134216575565643857232341160, −5.00467840285156766644711720864, −4.46403611617340679261276055296, −3.25094726918479187480187723121, −1.78835897153048611409238197888, 2.34714870930811456714803395176, 3.03833947422493891089299106565, 4.48834469662939847698210080683, 5.52394174977563707142887615001, 6.04144446616808067535662560202, 7.40408165820054030618382372317, 8.232968233715020234486085614552, 10.13233136880134929917442926415, 10.96028695330900032782537104321, 11.66837155068998834092733297893

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