Properties

Label 2-315-5.3-c2-0-7
Degree $2$
Conductor $315$
Sign $-0.337 - 0.941i$
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  + (−0.408 + 0.408i)2-s + 3.66i·4-s + (−0.563 − 4.96i)5-s + (−1.87 + 1.87i)7-s + (−3.13 − 3.13i)8-s + (2.25 + 1.79i)10-s + 6.25·11-s + (16.4 + 16.4i)13-s − 1.52i·14-s − 12.1·16-s + (−20.4 + 20.4i)17-s + 7.15i·19-s + (18.2 − 2.06i)20-s + (−2.55 + 2.55i)22-s + (12.0 + 12.0i)23-s + ⋯
L(s)  = 1  + (−0.204 + 0.204i)2-s + 0.916i·4-s + (−0.112 − 0.993i)5-s + (−0.267 + 0.267i)7-s + (−0.391 − 0.391i)8-s + (0.225 + 0.179i)10-s + 0.568·11-s + (1.26 + 1.26i)13-s − 0.109i·14-s − 0.756·16-s + (−1.20 + 1.20i)17-s + 0.376i·19-s + (0.910 − 0.103i)20-s + (−0.116 + 0.116i)22-s + (0.522 + 0.522i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.337 - 0.941i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.337 - 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.337 - 0.941i$
Analytic conductor: \(8.58312\)
Root analytic conductor: \(2.92969\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (253, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1),\ -0.337 - 0.941i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.646599 + 0.919183i\)
\(L(\frac12)\) \(\approx\) \(0.646599 + 0.919183i\)
\(L(2)\) 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 \( 1 + (0.563 + 4.96i)T \)
7 \( 1 + (1.87 - 1.87i)T \)
good2 \( 1 + (0.408 - 0.408i)T - 4iT^{2} \)
11 \( 1 - 6.25T + 121T^{2} \)
13 \( 1 + (-16.4 - 16.4i)T + 169iT^{2} \)
17 \( 1 + (20.4 - 20.4i)T - 289iT^{2} \)
19 \( 1 - 7.15iT - 361T^{2} \)
23 \( 1 + (-12.0 - 12.0i)T + 529iT^{2} \)
29 \( 1 - 18.1iT - 841T^{2} \)
31 \( 1 + 33.3T + 961T^{2} \)
37 \( 1 + (-18.8 + 18.8i)T - 1.36e3iT^{2} \)
41 \( 1 + 50.8T + 1.68e3T^{2} \)
43 \( 1 + (-53.3 - 53.3i)T + 1.84e3iT^{2} \)
47 \( 1 + (46.9 - 46.9i)T - 2.20e3iT^{2} \)
53 \( 1 + (-28.9 - 28.9i)T + 2.80e3iT^{2} \)
59 \( 1 + 10.0iT - 3.48e3T^{2} \)
61 \( 1 - 85.6T + 3.72e3T^{2} \)
67 \( 1 + (-11.9 + 11.9i)T - 4.48e3iT^{2} \)
71 \( 1 - 20.8T + 5.04e3T^{2} \)
73 \( 1 + (-35.2 - 35.2i)T + 5.32e3iT^{2} \)
79 \( 1 + 31.9iT - 6.24e3T^{2} \)
83 \( 1 + (6.49 + 6.49i)T + 6.88e3iT^{2} \)
89 \( 1 + 145. iT - 7.92e3T^{2} \)
97 \( 1 + (-31.5 + 31.5i)T - 9.40e3iT^{2} \)
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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

−11.71083845819236398321300150171, −11.08279331509977403389995119642, −9.336658080934924336390517731384, −8.900150202262979909033031675245, −8.174357557701255077298802697354, −6.88914258592321428790386310721, −6.00751922850485630584093794358, −4.37615802878165009206708995758, −3.64918856901947390406085961237, −1.68040395526263039112781574283, 0.58629105328736352818125597650, 2.41737234373800107699217794295, 3.70855729728168699628567684259, 5.25314595517578634928472709168, 6.38494335083819897217170219127, 7.02446488252828415466456202466, 8.482974154283414030909272734481, 9.441234703740568748765550068751, 10.39523059256439531670718323820, 11.01660542326587324516786734237

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