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Properties

Label 2-105-7.4-c3-0-0
Degree $2$
Conductor $105$
Sign $-0.181 - 0.983i$
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  + (−1.83 − 3.18i)2-s + (−1.5 + 2.59i)3-s + (−2.77 + 4.79i)4-s + (−2.5 − 4.33i)5-s + 11.0·6-s + (5.08 − 17.8i)7-s − 9.04·8-s + (−4.5 − 7.79i)9-s + (−9.19 + 15.9i)10-s + (−32.5 + 56.3i)11-s + (−8.31 − 14.3i)12-s − 6.87·13-s + (−66.1 + 16.5i)14-s + 15.0·15-s + (38.8 + 67.2i)16-s + (−34.0 + 58.9i)17-s + ⋯
L(s)  = 1  + (−0.650 − 1.12i)2-s + (−0.288 + 0.499i)3-s + (−0.346 + 0.599i)4-s + (−0.223 − 0.387i)5-s + 0.751·6-s + (0.274 − 0.961i)7-s − 0.399·8-s + (−0.166 − 0.288i)9-s + (−0.290 + 0.503i)10-s + (−0.891 + 1.54i)11-s + (−0.199 − 0.346i)12-s − 0.146·13-s + (−1.26 + 0.316i)14-s + 0.258·15-s + (0.606 + 1.05i)16-s + (−0.485 + 0.840i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.181 - 0.983i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.181 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $-0.181 - 0.983i$
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: \((2,\ 105,\ (\ :3/2),\ -0.181 - 0.983i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0256412 + 0.0307993i\)
\(L(\frac12)\) \(\approx\) \(0.0256412 + 0.0307993i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-5.08 + 17.8i)T \)
good2 \( 1 + (1.83 + 3.18i)T + (-4 + 6.92i)T^{2} \)
11 \( 1 + (32.5 - 56.3i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 6.87T + 2.19e3T^{2} \)
17 \( 1 + (34.0 - 58.9i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (11.7 + 20.3i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-2.74 - 4.76i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 138.T + 2.43e4T^{2} \)
31 \( 1 + (125. - 218. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (63.7 + 110. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 126.T + 6.89e4T^{2} \)
43 \( 1 - 91.5T + 7.95e4T^{2} \)
47 \( 1 + (284. + 492. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-295. + 511. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (254. - 441. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (130. + 226. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-461. + 799. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 519.T + 3.57e5T^{2} \)
73 \( 1 + (533. - 923. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-56.4 - 97.6i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 593.T + 5.71e5T^{2} \)
89 \( 1 + (-121. - 209. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 1.53e3T + 9.12e5T^{2} \)
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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

−13.08654927359734278448260269585, −12.34326681225941281322042587316, −11.15127404776035370039605885524, −10.42681412269498297663526340702, −9.693309676655131468918219333511, −8.447048996939304912909380080456, −7.06970402675030387713550286545, −5.08703108310779805543292292588, −3.81407876482468821749707271274, −1.85861528504102931978589823275, 0.02708510783300229298331801568, 2.80977033035475969259877980905, 5.43340991556799793277588941730, 6.22392962001499428523162111701, 7.52269040194476482079602091054, 8.298335744546814548426152848758, 9.300408737520202452490854611970, 10.98876790686901308413842780968, 11.80442534155154772644300745234, 13.09592338709510833900306485632

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