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Properties

Label 2-105-7.4-c3-0-5
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  + (2.68 + 4.64i)2-s + (−1.5 + 2.59i)3-s + (−10.3 + 17.9i)4-s + (2.5 + 4.33i)5-s − 16.0·6-s + (17.9 + 4.42i)7-s − 68.2·8-s + (−4.5 − 7.79i)9-s + (−13.4 + 23.2i)10-s + (27.8 − 48.2i)11-s + (−31.1 − 53.8i)12-s + 8.04·13-s + (27.6 + 95.3i)14-s − 15.0·15-s + (−100. − 173. i)16-s + (−48.9 + 84.7i)17-s + ⋯
L(s)  = 1  + (0.947 + 1.64i)2-s + (−0.288 + 0.499i)3-s + (−1.29 + 2.24i)4-s + (0.223 + 0.387i)5-s − 1.09·6-s + (0.970 + 0.239i)7-s − 3.01·8-s + (−0.166 − 0.288i)9-s + (−0.423 + 0.734i)10-s + (0.764 − 1.32i)11-s + (−0.748 − 1.29i)12-s + 0.171·13-s + (0.527 + 1.82i)14-s − 0.258·15-s + (−1.56 − 2.70i)16-s + (−0.697 + 1.20i)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: $\chi_{105} (46, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :3/2),\ -0.976 + 0.216i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.249215 - 2.27003i\)
\(L(\frac12)\) \(\approx\) \(0.249215 - 2.27003i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.5 - 2.59i)T \)
5 \( 1 + (-2.5 - 4.33i)T \)
7 \( 1 + (-17.9 - 4.42i)T \)
good2 \( 1 + (-2.68 - 4.64i)T + (-4 + 6.92i)T^{2} \)
11 \( 1 + (-27.8 + 48.2i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 8.04T + 2.19e3T^{2} \)
17 \( 1 + (48.9 - 84.7i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (31.5 + 54.6i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-69.4 - 120. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 229.T + 2.43e4T^{2} \)
31 \( 1 + (-10.7 + 18.6i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (27.9 + 48.4i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 256.T + 6.89e4T^{2} \)
43 \( 1 + 141.T + 7.95e4T^{2} \)
47 \( 1 + (-88.4 - 153. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-360. + 623. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (286. - 495. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (155. + 269. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-109. + 189. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 464.T + 3.57e5T^{2} \)
73 \( 1 + (-398. + 690. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (349. + 605. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 963.T + 5.71e5T^{2} \)
89 \( 1 + (25.7 + 44.5i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 1.60e3T + 9.12e5T^{2} \)
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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

−14.05207769324040988066201982787, −13.34813966217162846301514715696, −11.90522672188365442200159182240, −10.92055952845634577144279095810, −8.936941733447082869004626289708, −8.248965101409020136303742617005, −6.71984151644839542168956743082, −5.88619903536973070544774262361, −4.77515867893000584875773763433, −3.53760449866986078181581866039, 1.12925712937965342921742587316, 2.30499947698394323297720048032, 4.35543215036150837976215809969, 5.05554832144558460110712981789, 6.71780777874296692531497827706, 8.700426537763336421274441280148, 9.947990775814271618729366409893, 10.92959864963833406038121250027, 11.95975936076476193193439838928, 12.40031766410042730488865661110

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