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Properties

Label 2-105-35.4-c3-0-19
Degree $2$
Conductor $105$
Sign $0.600 + 0.799i$
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  + (3.00 − 1.73i)2-s + (2.59 + 1.5i)3-s + (2.01 − 3.49i)4-s + (1.37 − 11.0i)5-s + 10.4·6-s + (12.4 − 13.6i)7-s + 13.7i·8-s + (4.5 + 7.79i)9-s + (−15.1 − 35.7i)10-s + (13.2 − 23.0i)11-s + (10.4 − 6.05i)12-s + 8.46i·13-s + (13.7 − 62.7i)14-s + (20.2 − 26.7i)15-s + (39.9 + 69.2i)16-s + (−34.8 − 20.1i)17-s + ⋯
L(s)  = 1  + (1.06 − 0.613i)2-s + (0.499 + 0.288i)3-s + (0.252 − 0.437i)4-s + (0.122 − 0.992i)5-s + 0.708·6-s + (0.674 − 0.738i)7-s + 0.607i·8-s + (0.166 + 0.288i)9-s + (−0.478 − 1.12i)10-s + (0.364 − 0.630i)11-s + (0.252 − 0.145i)12-s + 0.180i·13-s + (0.263 − 1.19i)14-s + (0.347 − 0.460i)15-s + (0.624 + 1.08i)16-s + (−0.497 − 0.287i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.600 + 0.799i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.600 + 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.600 + 0.799i$
Analytic conductor: \(6.19520\)
Root analytic conductor: \(2.48901\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :3/2),\ 0.600 + 0.799i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.85326 - 1.42473i\)
\(L(\frac12)\) \(\approx\) \(2.85326 - 1.42473i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-1.37 + 11.0i)T \)
7 \( 1 + (-12.4 + 13.6i)T \)
good2 \( 1 + (-3.00 + 1.73i)T + (4 - 6.92i)T^{2} \)
11 \( 1 + (-13.2 + 23.0i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 8.46iT - 2.19e3T^{2} \)
17 \( 1 + (34.8 + 20.1i)T + (2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-50.5 - 87.5i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (123. - 71.5i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 80.4T + 2.43e4T^{2} \)
31 \( 1 + (-20.1 + 34.9i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (252. - 145. i)T + (2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 16.7T + 6.89e4T^{2} \)
43 \( 1 - 435. iT - 7.95e4T^{2} \)
47 \( 1 + (-278. + 160. i)T + (5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-187. - 108. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-246. + 427. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-392. - 680. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (629. + 363. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 546.T + 3.57e5T^{2} \)
73 \( 1 + (288. + 166. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (385. + 668. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 1.02e3iT - 5.71e5T^{2} \)
89 \( 1 + (-428. - 741. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 127. iT - 9.12e5T^{2} \)
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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

−13.41882969058091978158255803273, −12.12029184062512598955477910907, −11.38137309179120249144807018804, −10.06565037048903499661877110394, −8.728708461065020053224057682289, −7.80274030483692235938411792987, −5.67263931826782221050865082471, −4.51458096333257365183874944892, −3.62508839439167647096464107575, −1.67766153618278400669178142248, 2.32998586198599626735370323675, 3.93150805087323526487065653759, 5.38326112215685159551760636507, 6.59374340282180351846961783335, 7.47772489151309945451160647915, 8.967939778877676198738285081506, 10.26111016960032832223865028598, 11.66796342971075877750528816284, 12.64584141130560464141802470445, 13.83004023149428869903846585514

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