Properties

Label 2-105-105.104-c3-0-19
Degree $2$
Conductor $105$
Sign $0.781 - 0.623i$
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  − 1.22·2-s + (4.91 + 1.67i)3-s − 6.50·4-s + (9.89 + 5.21i)5-s + (−6.01 − 2.04i)6-s + (1.55 − 18.4i)7-s + 17.7·8-s + (21.4 + 16.4i)9-s + (−12.0 − 6.37i)10-s + 54.5i·11-s + (−32.0 − 10.8i)12-s + 24.4·13-s + (−1.90 + 22.5i)14-s + (39.9 + 42.1i)15-s + 30.3·16-s − 36.0i·17-s + ⋯
L(s)  = 1  − 0.432·2-s + (0.946 + 0.322i)3-s − 0.813·4-s + (0.884 + 0.466i)5-s + (−0.409 − 0.139i)6-s + (0.0841 − 0.996i)7-s + 0.783·8-s + (0.792 + 0.609i)9-s + (−0.382 − 0.201i)10-s + 1.49i·11-s + (−0.769 − 0.261i)12-s + 0.522·13-s + (−0.0363 + 0.430i)14-s + (0.687 + 0.726i)15-s + 0.474·16-s − 0.513i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.781 - 0.623i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.781 - 0.623i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.781 - 0.623i$
Analytic conductor: \(6.19520\)
Root analytic conductor: \(2.48901\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (104, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :3/2),\ 0.781 - 0.623i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.61831 + 0.566634i\)
\(L(\frac12)\) \(\approx\) \(1.61831 + 0.566634i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-4.91 - 1.67i)T \)
5 \( 1 + (-9.89 - 5.21i)T \)
7 \( 1 + (-1.55 + 18.4i)T \)
good2 \( 1 + 1.22T + 8T^{2} \)
11 \( 1 - 54.5iT - 1.33e3T^{2} \)
13 \( 1 - 24.4T + 2.19e3T^{2} \)
17 \( 1 + 36.0iT - 4.91e3T^{2} \)
19 \( 1 - 99.5iT - 6.85e3T^{2} \)
23 \( 1 - 97.3T + 1.21e4T^{2} \)
29 \( 1 - 14.1iT - 2.43e4T^{2} \)
31 \( 1 + 186. iT - 2.97e4T^{2} \)
37 \( 1 + 199. iT - 5.06e4T^{2} \)
41 \( 1 + 313.T + 6.89e4T^{2} \)
43 \( 1 - 30.0iT - 7.95e4T^{2} \)
47 \( 1 + 514. iT - 1.03e5T^{2} \)
53 \( 1 + 451.T + 1.48e5T^{2} \)
59 \( 1 + 355.T + 2.05e5T^{2} \)
61 \( 1 + 656. iT - 2.26e5T^{2} \)
67 \( 1 - 73.7iT - 3.00e5T^{2} \)
71 \( 1 - 611. iT - 3.57e5T^{2} \)
73 \( 1 - 424.T + 3.89e5T^{2} \)
79 \( 1 - 814.T + 4.93e5T^{2} \)
83 \( 1 + 585. iT - 5.71e5T^{2} \)
89 \( 1 - 732.T + 7.04e5T^{2} \)
97 \( 1 + 272.T + 9.12e5T^{2} \)
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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

−13.58411627939539874187977484241, −12.80234100938101621664638255138, −10.69584098849344428505936190329, −9.923857034439032355406905532466, −9.346112485630464783520133847915, −7.981586072065595982608853129708, −7.01211650695401529645961783531, −4.94517334004510804562314437463, −3.71570896726810436247453466698, −1.72265005840392622990174503772, 1.25433630125440740354803222747, 3.07330623881337971534832119007, 4.97575871167518981227685825006, 6.32986459985884526298470578160, 8.246327608336585309247309575656, 8.813404403397637143442464135887, 9.422107236235323231785142055773, 10.82171050153827116273146280394, 12.49491125649768954576150663551, 13.42596526408557702791917929088

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