Properties

Label 2-105-21.5-c3-0-12
Degree $2$
Conductor $105$
Sign $0.213 - 0.976i$
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  + (4.69 + 2.70i)2-s + (−2.85 − 4.34i)3-s + (10.6 + 18.4i)4-s + (−2.5 + 4.33i)5-s + (−1.61 − 28.0i)6-s + (14.5 + 11.4i)7-s + 72.2i·8-s + (−10.7 + 24.7i)9-s + (−23.4 + 13.5i)10-s + (38.4 − 22.2i)11-s + (49.8 − 99.0i)12-s − 22.9i·13-s + (36.9 + 93.2i)14-s + (25.9 − 1.49i)15-s + (−110. + 191. i)16-s + (−32.3 − 55.9i)17-s + ⋯
L(s)  = 1  + (1.65 + 0.957i)2-s + (−0.548 − 0.835i)3-s + (1.33 + 2.30i)4-s + (−0.223 + 0.387i)5-s + (−0.110 − 1.91i)6-s + (0.784 + 0.620i)7-s + 3.19i·8-s + (−0.397 + 0.917i)9-s + (−0.741 + 0.428i)10-s + (1.05 − 0.608i)11-s + (1.19 − 2.38i)12-s − 0.489i·13-s + (0.706 + 1.78i)14-s + (0.446 − 0.0256i)15-s + (−1.72 + 2.98i)16-s + (−0.461 − 0.798i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.213 - 0.976i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.213 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.213 - 0.976i$
Analytic conductor: \(6.19520\)
Root analytic conductor: \(2.48901\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :3/2),\ 0.213 - 0.976i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.61735 + 2.10707i\)
\(L(\frac12)\) \(\approx\) \(2.61735 + 2.10707i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.85 + 4.34i)T \)
5 \( 1 + (2.5 - 4.33i)T \)
7 \( 1 + (-14.5 - 11.4i)T \)
good2 \( 1 + (-4.69 - 2.70i)T + (4 + 6.92i)T^{2} \)
11 \( 1 + (-38.4 + 22.2i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 22.9iT - 2.19e3T^{2} \)
17 \( 1 + (32.3 + 55.9i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (15.6 + 9.06i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (76.2 + 44.0i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 308. iT - 2.43e4T^{2} \)
31 \( 1 + (82.5 - 47.6i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (83.8 - 145. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 271.T + 6.89e4T^{2} \)
43 \( 1 - 225.T + 7.95e4T^{2} \)
47 \( 1 + (31.1 - 54.0i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-19.9 + 11.5i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (41.0 + 71.1i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-424. - 245. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (278. + 482. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 669. iT - 3.57e5T^{2} \)
73 \( 1 + (184. - 106. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-288. + 499. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 1.28e3T + 5.71e5T^{2} \)
89 \( 1 + (511. - 885. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 472. iT - 9.12e5T^{2} \)
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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

−13.73432130273189751934801103855, −12.58719035348878210096990840112, −11.71691724868050156129725165946, −11.23274984364801981330470922836, −8.445165640302800972107577885155, −7.50829763602007501939347600249, −6.40642583372994182445955050180, −5.63363973834220861636923556176, −4.32417765924499968907828137496, −2.53748040804935555833792778315, 1.53815734327878833548358909599, 3.86586316067911437505668995622, 4.37721423273777559814496595626, 5.53313150439656458880373812364, 6.83631096228030284134632075449, 9.204633337794772716295116102575, 10.46017284955191537700452092987, 11.18498063050927261471628798773, 11.99690089750345650719216483954, 12.81540641198242073402189817720

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