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Properties

Label 2-105-35.9-c3-0-15
Degree $2$
Conductor $105$
Sign $0.615 - 0.788i$
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  + (4.10 + 2.37i)2-s + (2.59 − 1.5i)3-s + (7.23 + 12.5i)4-s + (10.7 − 3.12i)5-s + 14.2·6-s + (−15.6 + 9.85i)7-s + 30.7i·8-s + (4.5 − 7.79i)9-s + (51.4 + 12.6i)10-s + (−18.3 − 31.7i)11-s + (37.6 + 21.7i)12-s + 11.1i·13-s + (−87.7 + 3.28i)14-s + (23.2 − 24.2i)15-s + (−14.9 + 25.8i)16-s + (31.6 − 18.2i)17-s + ⋯
L(s)  = 1  + (1.45 + 0.838i)2-s + (0.499 − 0.288i)3-s + (0.904 + 1.56i)4-s + (0.960 − 0.279i)5-s + 0.967·6-s + (−0.846 + 0.532i)7-s + 1.35i·8-s + (0.166 − 0.288i)9-s + (1.62 + 0.399i)10-s + (−0.502 − 0.871i)11-s + (0.904 + 0.522i)12-s + 0.237i·13-s + (−1.67 + 0.0627i)14-s + (0.399 − 0.416i)15-s + (−0.232 + 0.403i)16-s + (0.451 − 0.260i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.615 - 0.788i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.615 - 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.615 - 0.788i$
Analytic conductor: \(6.19520\)
Root analytic conductor: \(2.48901\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :3/2),\ 0.615 - 0.788i)\)

Particular Values

\(L(2)\) \(\approx\) \(3.54315 + 1.72819i\)
\(L(\frac12)\) \(\approx\) \(3.54315 + 1.72819i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.59 + 1.5i)T \)
5 \( 1 + (-10.7 + 3.12i)T \)
7 \( 1 + (15.6 - 9.85i)T \)
good2 \( 1 + (-4.10 - 2.37i)T + (4 + 6.92i)T^{2} \)
11 \( 1 + (18.3 + 31.7i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 11.1iT - 2.19e3T^{2} \)
17 \( 1 + (-31.6 + 18.2i)T + (2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (75.6 - 131. i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (59.1 + 34.1i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 167.T + 2.43e4T^{2} \)
31 \( 1 + (130. + 226. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-166. - 95.8i)T + (2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 142.T + 6.89e4T^{2} \)
43 \( 1 + 122. iT - 7.95e4T^{2} \)
47 \( 1 + (-259. - 149. i)T + (5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-362. + 209. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-214. - 371. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (390. - 676. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (794. - 458. i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 263.T + 3.57e5T^{2} \)
73 \( 1 + (276. - 159. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-369. + 639. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 182. iT - 5.71e5T^{2} \)
89 \( 1 + (446. - 772. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 311. iT - 9.12e5T^{2} \)
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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

−13.39560476069731460432197267094, −12.94432529471044237032584433370, −11.99519926230724183025149797902, −10.13409923508559533649636703290, −8.864388135069055152449264201781, −7.57585486489663320570702001214, −6.06155642876315727326506892144, −5.75921933942768188170738081759, −3.90928783567699579061225768787, −2.51823042832692003452396675682, 2.12305438248494461827242362062, 3.25642384268054806582696100502, 4.59612906507335550343919036128, 5.85533657866459750196143789881, 7.14811864422272202484982161300, 9.274946458719384195079376626381, 10.27030542453634701437401166305, 10.93032195247420646289052521433, 12.62957412298416075258822928506, 13.08345748758228683263398840460

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