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Properties

Label 2-105-7.4-c3-0-13
Degree $2$
Conductor $105$
Sign $-0.677 + 0.735i$
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  + (−0.684 − 1.18i)2-s + (1.5 − 2.59i)3-s + (3.06 − 5.30i)4-s + (−2.5 − 4.33i)5-s − 4.10·6-s + (17.2 − 6.62i)7-s − 19.3·8-s + (−4.5 − 7.79i)9-s + (−3.42 + 5.92i)10-s + (−4.53 + 7.85i)11-s + (−9.19 − 15.9i)12-s − 2.69·13-s + (−19.6 − 15.9i)14-s − 15.0·15-s + (−11.2 − 19.5i)16-s + (−7.75 + 13.4i)17-s + ⋯
L(s)  = 1  + (−0.241 − 0.418i)2-s + (0.288 − 0.499i)3-s + (0.383 − 0.663i)4-s + (−0.223 − 0.387i)5-s − 0.279·6-s + (0.933 − 0.357i)7-s − 0.854·8-s + (−0.166 − 0.288i)9-s + (−0.108 + 0.187i)10-s + (−0.124 + 0.215i)11-s + (−0.221 − 0.383i)12-s − 0.0575·13-s + (−0.375 − 0.304i)14-s − 0.258·15-s + (−0.176 − 0.305i)16-s + (−0.110 + 0.191i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.677 + 0.735i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.677 + 0.735i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $-0.677 + 0.735i$
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: \((2,\ 105,\ (\ :3/2),\ -0.677 + 0.735i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.624044 - 1.42347i\)
\(L(\frac12)\) \(\approx\) \(0.624044 - 1.42347i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-17.2 + 6.62i)T \)
good2 \( 1 + (0.684 + 1.18i)T + (-4 + 6.92i)T^{2} \)
11 \( 1 + (4.53 - 7.85i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 2.69T + 2.19e3T^{2} \)
17 \( 1 + (7.75 - 13.4i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (23.2 + 40.2i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (23.7 + 41.1i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 8.21T + 2.43e4T^{2} \)
31 \( 1 + (-114. + 197. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-167. - 289. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 35.3T + 6.89e4T^{2} \)
43 \( 1 - 510.T + 7.95e4T^{2} \)
47 \( 1 + (-272. - 472. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-2.16 + 3.75i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-395. + 685. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-107. - 186. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (201. - 348. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 328.T + 3.57e5T^{2} \)
73 \( 1 + (-261. + 453. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-65.3 - 113. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 507.T + 5.71e5T^{2} \)
89 \( 1 + (282. + 490. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 1.43e3T + 9.12e5T^{2} \)
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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

−12.78393225624308155529888684446, −11.68950908687045244184335289087, −10.91993899200166801075386632452, −9.719806148182179878806465769114, −8.517885519934776256597331582630, −7.42446425275668150782998200118, −6.04418333974008959155377986857, −4.52069505487979650340555968236, −2.37831550533182383578903319775, −0.946342798995600140560875231429, 2.53022254681198202175888219895, 4.02901928757753137241814039211, 5.70216541309324782862983543151, 7.23520918014834653979159136131, 8.166848257997470596741944460104, 9.044603658198961683127199945374, 10.58040048602841740563971815089, 11.51392735161773626430496653051, 12.44620639427552418973453718546, 13.95567601173483224386180347160

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