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Properties

Label 2-117-13.12-c3-0-1
Degree $2$
Conductor $117$
Sign $-0.719 + 0.694i$
Analytic cond. $6.90322$
Root an. cond. $2.62739$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.54i·2-s − 12.6·4-s + 12.9i·5-s + 16.7i·7-s − 21.2i·8-s − 58.7·10-s − 24.9i·11-s + (33.7 − 32.5i)13-s − 76.0·14-s − 4.67·16-s − 134.·17-s + 14.9i·19-s − 163. i·20-s + 113.·22-s + 72·23-s + ⋯
L(s)  = 1  + 1.60i·2-s − 1.58·4-s + 1.15i·5-s + 0.903i·7-s − 0.940i·8-s − 1.85·10-s − 0.683i·11-s + (0.719 − 0.694i)13-s − 1.45·14-s − 0.0731·16-s − 1.91·17-s + 0.180i·19-s − 1.83i·20-s + 1.09·22-s + 0.652·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.719 + 0.694i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.719 + 0.694i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $-0.719 + 0.694i$
Analytic conductor: \(6.90322\)
Root analytic conductor: \(2.62739\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :3/2),\ -0.719 + 0.694i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.444339 - 1.09968i\)
\(L(\frac12)\) \(\approx\) \(0.444339 - 1.09968i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 + (-33.7 + 32.5i)T \)
good2 \( 1 - 4.54iT - 8T^{2} \)
5 \( 1 - 12.9iT - 125T^{2} \)
7 \( 1 - 16.7iT - 343T^{2} \)
11 \( 1 + 24.9iT - 1.33e3T^{2} \)
17 \( 1 + 134.T + 4.91e3T^{2} \)
19 \( 1 - 14.9iT - 6.85e3T^{2} \)
23 \( 1 - 72T + 1.21e4T^{2} \)
29 \( 1 - 206.T + 2.43e4T^{2} \)
31 \( 1 - 249. iT - 2.97e4T^{2} \)
37 \( 1 - 293. iT - 5.06e4T^{2} \)
41 \( 1 - 250. iT - 6.89e4T^{2} \)
43 \( 1 + 432.T + 7.95e4T^{2} \)
47 \( 1 - 159. iT - 1.03e5T^{2} \)
53 \( 1 - 194.T + 1.48e5T^{2} \)
59 \( 1 + 232. iT - 2.05e5T^{2} \)
61 \( 1 + 185.T + 2.26e5T^{2} \)
67 \( 1 - 39.4iT - 3.00e5T^{2} \)
71 \( 1 - 920. iT - 3.57e5T^{2} \)
73 \( 1 + 549. iT - 3.89e5T^{2} \)
79 \( 1 - 933.T + 4.93e5T^{2} \)
83 \( 1 + 1.09e3iT - 5.71e5T^{2} \)
89 \( 1 - 532. iT - 7.04e5T^{2} \)
97 \( 1 - 362. iT - 9.12e5T^{2} \)
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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

−13.97865230901240122326860865825, −13.17096814115004453528770512376, −11.51432987397632014511280927319, −10.52348459487323165198100093751, −8.876749095230373174791808701947, −8.256627592157154795934583097301, −6.74361625982959498936518871315, −6.28686298370188915442704540550, −4.95661802914526892062521586123, −2.96765224285900746052429781655, 0.64492288484619623777322623743, 2.03677738579629372244051288228, 4.02443129768475471476256241117, 4.70176294154618383515049409176, 6.85628832495656741693293592618, 8.650307644997908360771824447395, 9.343723002851688323840401593847, 10.53212125852560436262777164633, 11.34895504646181763608019454364, 12.34722720571570713282008315208

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