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Properties

Label 2-105-35.4-c3-0-7
Degree $2$
Conductor $105$
Sign $0.933 - 0.358i$
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 + (−10.2 − 4.35i)5-s + 10.4·6-s + (−12.4 + 13.6i)7-s − 13.7i·8-s + (4.5 + 7.79i)9-s + (38.4 − 4.76i)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 + (−0.920 − 0.389i)5-s + 0.708·6-s + (−0.674 + 0.738i)7-s − 0.607i·8-s + (0.166 + 0.288i)9-s + (1.21 − 0.150i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.933 - 0.358i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.933 - 0.358i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.933 - 0.358i$
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: \((2,\ 105,\ (\ :3/2),\ 0.933 - 0.358i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.500417 + 0.0929181i\)
\(L(\frac12)\) \(\approx\) \(0.500417 + 0.0929181i\)
\(L(\frac{5}{2})\) 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 \( 1 + (10.2 + 4.35i)T \)
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

−12.97839634358511121358257326143, −12.32040728476716052578544285675, −11.21611501379789333317969695365, −9.841146011003775168116388978311, −8.776657341154866510066742474512, −7.938738926496319848405162069429, −6.81442938340145321581801730343, −5.61856016130982863320794561328, −3.62402073739506235439765610510, −0.72009584327367952131387713479, 0.800506542117276298963137066731, 3.21220153084981794686853918343, 4.80372301376681817413895872864, 6.78300744762435723009033950664, 7.74574715798250042695225191082, 9.295852796233365771104898637425, 9.969468516650495807178750280792, 11.12939175841224620394774867323, 11.59097904786018244868996038706, 12.91617103744662141419559985366

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