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Properties

Label 2-105-35.9-c3-0-13
Degree $2$
Conductor $105$
Sign $-0.976 + 0.216i$
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  + (−3.89 − 2.24i)2-s + (2.59 − 1.5i)3-s + (6.10 + 10.5i)4-s + (−0.533 + 11.1i)5-s − 13.4·6-s + (−3.71 − 18.1i)7-s − 18.9i·8-s + (4.5 − 7.79i)9-s + (27.1 − 42.2i)10-s + (−11.2 − 19.4i)11-s + (31.7 + 18.3i)12-s − 70.4i·13-s + (−26.3 + 78.9i)14-s + (15.3 + 29.8i)15-s + (6.30 − 10.9i)16-s + (−52.3 + 30.2i)17-s + ⋯
L(s)  = 1  + (−1.37 − 0.794i)2-s + (0.499 − 0.288i)3-s + (0.763 + 1.32i)4-s + (−0.0476 + 0.998i)5-s − 0.917·6-s + (−0.200 − 0.979i)7-s − 0.836i·8-s + (0.166 − 0.288i)9-s + (0.859 − 1.33i)10-s + (−0.307 − 0.532i)11-s + (0.763 + 0.440i)12-s − 1.50i·13-s + (−0.502 + 1.50i)14-s + (0.264 + 0.513i)15-s + (0.0985 − 0.170i)16-s + (−0.747 + 0.431i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.976 + 0.216i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.976 + 0.216i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $-0.976 + 0.216i$
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: \((2,\ 105,\ (\ :3/2),\ -0.976 + 0.216i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0558864 - 0.511043i\)
\(L(\frac12)\) \(\approx\) \(0.0558864 - 0.511043i\)
\(L(\frac{5}{2})\) 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 \( 1 + (0.533 - 11.1i)T \)
7 \( 1 + (3.71 + 18.1i)T \)
good2 \( 1 + (3.89 + 2.24i)T + (4 + 6.92i)T^{2} \)
11 \( 1 + (11.2 + 19.4i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 70.4iT - 2.19e3T^{2} \)
17 \( 1 + (52.3 - 30.2i)T + (2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (18.2 - 31.5i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (169. + 98.0i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 144.T + 2.43e4T^{2} \)
31 \( 1 + (88.3 + 153. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-190. - 109. i)T + (2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 157.T + 6.89e4T^{2} \)
43 \( 1 + 99.6iT - 7.95e4T^{2} \)
47 \( 1 + (372. + 214. i)T + (5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-360. + 208. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (385. + 668. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (46.2 - 80.0i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-23.7 + 13.6i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 388.T + 3.57e5T^{2} \)
73 \( 1 + (-200. + 115. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (58.7 - 101. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 1.07e3iT - 5.71e5T^{2} \)
89 \( 1 + (93.5 - 162. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 415. iT - 9.12e5T^{2} \)
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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

−12.64247064064999808396980813444, −11.24953880956212045923974344868, −10.43521698562146810088550891466, −9.907494966470214934233067465707, −8.264762048736592360220956683717, −7.76554593826282808416329641901, −6.37697599636279316547642366173, −3.58784575376348900311042336511, −2.36506933473743784271393035602, −0.40469801481149778547341323551, 1.91840810636909424136291513780, 4.52302636452947791063533608604, 6.10145862200540706951969385573, 7.45444715244409578204581979347, 8.607907081271661601703672593058, 9.170517948212627588696619787887, 9.883104393131937217746688085605, 11.52642704599274516795200788934, 12.68910207566785739842695873429, 14.01065431169163276082631054456

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