Properties

Label 2-105-35.9-c3-0-18
Degree $2$
Conductor $105$
Sign $-0.114 + 0.993i$
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  + (2.21 + 1.27i)2-s + (−2.59 + 1.5i)3-s + (−0.732 − 1.26i)4-s + (−11.1 − 0.184i)5-s − 7.66·6-s + (2.98 − 18.2i)7-s − 24.1i·8-s + (4.5 − 7.79i)9-s + (−24.5 − 14.6i)10-s + (0.664 + 1.15i)11-s + (3.80 + 2.19i)12-s − 46.0i·13-s + (29.9 − 36.6i)14-s + (29.3 − 16.2i)15-s + (25.0 − 43.4i)16-s + (−77.3 + 44.6i)17-s + ⋯
L(s)  = 1  + (0.782 + 0.451i)2-s + (−0.499 + 0.288i)3-s + (−0.0916 − 0.158i)4-s + (−0.999 − 0.0164i)5-s − 0.521·6-s + (0.161 − 0.986i)7-s − 1.06i·8-s + (0.166 − 0.288i)9-s + (−0.775 − 0.464i)10-s + (0.0182 + 0.0315i)11-s + (0.0916 + 0.0528i)12-s − 0.982i·13-s + (0.572 − 0.699i)14-s + (0.504 − 0.280i)15-s + (0.391 − 0.678i)16-s + (−1.10 + 0.636i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.114 + 0.993i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.114 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $-0.114 + 0.993i$
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: \((2,\ 105,\ (\ :3/2),\ -0.114 + 0.993i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.662947 - 0.744028i\)
\(L(\frac12)\) \(\approx\) \(0.662947 - 0.744028i\)
\(L(\frac{5}{2})\) 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 \( 1 + (11.1 + 0.184i)T \)
7 \( 1 + (-2.98 + 18.2i)T \)
good2 \( 1 + (-2.21 - 1.27i)T + (4 + 6.92i)T^{2} \)
11 \( 1 + (-0.664 - 1.15i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 46.0iT - 2.19e3T^{2} \)
17 \( 1 + (77.3 - 44.6i)T + (2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (27.1 - 47.1i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (132. + 76.4i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 154.T + 2.43e4T^{2} \)
31 \( 1 + (57.7 + 100. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-112. - 64.8i)T + (2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 128.T + 6.89e4T^{2} \)
43 \( 1 + 428. iT - 7.95e4T^{2} \)
47 \( 1 + (-487. - 281. i)T + (5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-77.5 + 44.7i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-292. - 506. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-170. + 296. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-83.8 + 48.4i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 673.T + 3.57e5T^{2} \)
73 \( 1 + (494. - 285. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-298. + 517. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 801. iT - 5.71e5T^{2} \)
89 \( 1 + (-69.1 + 119. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 1.52e3iT - 9.12e5T^{2} \)
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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

−13.05260822018909097108697357177, −12.16459629498189449644562811189, −10.79780260888492139390981103937, −10.16218132515843977796278428053, −8.372191682219281380012204450385, −7.12635092220859813869756373956, −5.98769488871264229246549773995, −4.54195256060199429506193846858, −3.86089134200365741747231172423, −0.45880040025417417934873645555, 2.41992440369291059567539532549, 4.10002083936823364173381625827, 5.11422685051282230942026957792, 6.67018303673240949851036621067, 8.085567305185503009664391551232, 9.078072052562648282556522215985, 11.06358848598910289930674139992, 11.73132551188122044119332407294, 12.26969508261966665936023594726, 13.35453318113734390623349779169

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