Properties

Label 2-105-5.4-c3-0-6
Degree $2$
Conductor $105$
Sign $0.134 - 0.990i$
Analytic cond. $6.19520$
Root an. cond. $2.48901$
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  + 2.20i·2-s − 3i·3-s + 3.13·4-s + (−1.50 + 11.0i)5-s + 6.61·6-s − 7i·7-s + 24.5i·8-s − 9·9-s + (−24.4 − 3.31i)10-s + 56.2·11-s − 9.39i·12-s + 38.9i·13-s + 15.4·14-s + (33.2 + 4.50i)15-s − 29.1·16-s + 119. i·17-s + ⋯
L(s)  = 1  + 0.780i·2-s − 0.577i·3-s + 0.391·4-s + (−0.134 + 0.990i)5-s + 0.450·6-s − 0.377i·7-s + 1.08i·8-s − 0.333·9-s + (−0.773 − 0.104i)10-s + 1.54·11-s − 0.225i·12-s + 0.829i·13-s + 0.294·14-s + (0.572 + 0.0774i)15-s − 0.455·16-s + 1.70i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.134 - 0.990i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.134 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.134 - 0.990i$
Analytic conductor: \(6.19520\)
Root analytic conductor: \(2.48901\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :3/2),\ 0.134 - 0.990i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.36379 + 1.19154i\)
\(L(\frac12)\) \(\approx\) \(1.36379 + 1.19154i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 3iT \)
5 \( 1 + (1.50 - 11.0i)T \)
7 \( 1 + 7iT \)
good2 \( 1 - 2.20iT - 8T^{2} \)
11 \( 1 - 56.2T + 1.33e3T^{2} \)
13 \( 1 - 38.9iT - 2.19e3T^{2} \)
17 \( 1 - 119. iT - 4.91e3T^{2} \)
19 \( 1 - 13.0T + 6.85e3T^{2} \)
23 \( 1 + 130. iT - 1.21e4T^{2} \)
29 \( 1 + 77.9T + 2.43e4T^{2} \)
31 \( 1 - 61.0T + 2.97e4T^{2} \)
37 \( 1 + 167. iT - 5.06e4T^{2} \)
41 \( 1 - 436.T + 6.89e4T^{2} \)
43 \( 1 + 393. iT - 7.95e4T^{2} \)
47 \( 1 + 365. iT - 1.03e5T^{2} \)
53 \( 1 + 282. iT - 1.48e5T^{2} \)
59 \( 1 + 414.T + 2.05e5T^{2} \)
61 \( 1 + 563.T + 2.26e5T^{2} \)
67 \( 1 + 395. iT - 3.00e5T^{2} \)
71 \( 1 - 103.T + 3.57e5T^{2} \)
73 \( 1 + 128. iT - 3.89e5T^{2} \)
79 \( 1 - 641.T + 4.93e5T^{2} \)
83 \( 1 - 512. iT - 5.71e5T^{2} \)
89 \( 1 + 1.22e3T + 7.04e5T^{2} \)
97 \( 1 + 186. iT - 9.12e5T^{2} \)
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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

−14.02189073940758251930027127563, −12.38155404507160667734678521625, −11.42356877787083057219151931211, −10.55688651325750598636282450830, −8.866736467249277534730577243677, −7.63541582862902403401601979681, −6.66587149583901764800399770219, −6.16051543027244869812668382390, −3.89359918634416439519060874551, −1.97318899501077797207547243183, 1.13786928987769514667714563043, 3.09883428944997569208472953217, 4.48577396528645633129518616264, 5.94538851466182320962010279464, 7.58097259092465391854607605973, 9.223209671454821813758374708940, 9.636381564175456747743064655069, 11.23795219072681947130069761421, 11.80032758057421074149681810010, 12.70596197612400288394485135070

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