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Properties

Label 2-105-21.5-c3-0-20
Degree $2$
Conductor $105$
Sign $-0.571 + 0.820i$
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  + (−3.64 − 2.10i)2-s + (4.12 − 3.16i)3-s + (4.87 + 8.44i)4-s + (2.5 − 4.33i)5-s + (−21.7 + 2.86i)6-s + (16.7 − 7.79i)7-s − 7.40i·8-s + (6.97 − 26.0i)9-s + (−18.2 + 10.5i)10-s + (50.2 − 28.9i)11-s + (46.8 + 19.3i)12-s + 47.9i·13-s + (−77.7 − 6.94i)14-s + (−3.39 − 25.7i)15-s + (23.4 − 40.5i)16-s + (11.9 + 20.6i)17-s + ⋯
L(s)  = 1  + (−1.29 − 0.744i)2-s + (0.793 − 0.609i)3-s + (0.609 + 1.05i)4-s + (0.223 − 0.387i)5-s + (−1.47 + 0.194i)6-s + (0.907 − 0.420i)7-s − 0.327i·8-s + (0.258 − 0.966i)9-s + (−0.577 + 0.333i)10-s + (1.37 − 0.794i)11-s + (1.12 + 0.466i)12-s + 1.02i·13-s + (−1.48 − 0.132i)14-s + (−0.0585 − 0.443i)15-s + (0.366 − 0.634i)16-s + (0.169 + 0.294i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.571 + 0.820i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.571 + 0.820i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $-0.571 + 0.820i$
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: \((2,\ 105,\ (\ :3/2),\ -0.571 + 0.820i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.559269 - 1.07096i\)
\(L(\frac12)\) \(\approx\) \(0.559269 - 1.07096i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-4.12 + 3.16i)T \)
5 \( 1 + (-2.5 + 4.33i)T \)
7 \( 1 + (-16.7 + 7.79i)T \)
good2 \( 1 + (3.64 + 2.10i)T + (4 + 6.92i)T^{2} \)
11 \( 1 + (-50.2 + 28.9i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 47.9iT - 2.19e3T^{2} \)
17 \( 1 + (-11.9 - 20.6i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (116. + 67.2i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (84.9 + 49.0i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 57.6iT - 2.43e4T^{2} \)
31 \( 1 + (124. - 71.6i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (0.379 - 0.657i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 518.T + 6.89e4T^{2} \)
43 \( 1 - 201.T + 7.95e4T^{2} \)
47 \( 1 + (140. - 243. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (168. - 97.2i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (244. + 423. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-159. - 91.8i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-452. - 783. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 524. iT - 3.57e5T^{2} \)
73 \( 1 + (110. - 64.0i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-120. + 208. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 607.T + 5.71e5T^{2} \)
89 \( 1 + (436. - 756. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 1.29e3iT - 9.12e5T^{2} \)
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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

−12.67841519582418690684910265042, −11.62940432302983500482484753295, −10.78832908369929056575269927825, −9.264943788171403443000810179524, −8.820876148775789459959283670316, −7.86126883212294314097097302128, −6.50340653002333960466145837430, −4.09215101191931777337172018485, −2.10747871118300391897429092767, −1.05160110819986287549558243172, 1.89628516294499720470682937861, 4.07821733113517398577151693445, 5.93760231696721747460150645248, 7.44190727271725579603556584490, 8.238529394333828269721737238113, 9.220046476702378954176528151851, 10.00939149167594831395889348716, 11.02104979710239699152159557848, 12.59969589982577280330221901135, 14.28592789536995815242291675813

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