Properties

Label 2-105-21.17-c3-0-7
Degree $2$
Conductor $105$
Sign $-0.840 - 0.541i$
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  + (−4.57 + 2.63i)2-s + (4.83 + 1.91i)3-s + (9.93 − 17.2i)4-s + (−2.5 − 4.33i)5-s + (−27.1 + 3.99i)6-s + (−17.5 − 5.84i)7-s + 62.6i·8-s + (19.6 + 18.4i)9-s + (22.8 + 13.1i)10-s + (19.7 + 11.4i)11-s + (80.9 − 64.1i)12-s + 81.4i·13-s + (95.7 − 19.6i)14-s + (−3.78 − 25.7i)15-s + (−85.9 − 148. i)16-s + (−61.9 + 107. i)17-s + ⋯
L(s)  = 1  + (−1.61 + 0.933i)2-s + (0.929 + 0.368i)3-s + (1.24 − 2.15i)4-s + (−0.223 − 0.387i)5-s + (−1.84 + 0.272i)6-s + (−0.948 − 0.315i)7-s + 2.77i·8-s + (0.728 + 0.685i)9-s + (0.722 + 0.417i)10-s + (0.542 + 0.313i)11-s + (1.94 − 1.54i)12-s + 1.73i·13-s + (1.82 − 0.375i)14-s + (−0.0651 − 0.442i)15-s + (−1.34 − 2.32i)16-s + (−0.884 + 1.53i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.840 - 0.541i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.840 - 0.541i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $-0.840 - 0.541i$
Analytic conductor: \(6.19520\)
Root analytic conductor: \(2.48901\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :3/2),\ -0.840 - 0.541i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.193245 + 0.657453i\)
\(L(\frac12)\) \(\approx\) \(0.193245 + 0.657453i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-4.83 - 1.91i)T \)
5 \( 1 + (2.5 + 4.33i)T \)
7 \( 1 + (17.5 + 5.84i)T \)
good2 \( 1 + (4.57 - 2.63i)T + (4 - 6.92i)T^{2} \)
11 \( 1 + (-19.7 - 11.4i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 81.4iT - 2.19e3T^{2} \)
17 \( 1 + (61.9 - 107. i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (40.3 - 23.2i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-98.1 + 56.6i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 80.2iT - 2.43e4T^{2} \)
31 \( 1 + (15.7 + 9.06i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (49.5 + 85.7i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 142.T + 6.89e4T^{2} \)
43 \( 1 - 184.T + 7.95e4T^{2} \)
47 \( 1 + (-60.6 - 105. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (200. + 115. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (3.91 - 6.77i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-158. + 91.6i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (141. - 244. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 521. iT - 3.57e5T^{2} \)
73 \( 1 + (-77.7 - 44.8i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (298. + 517. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 249.T + 5.71e5T^{2} \)
89 \( 1 + (-147. - 256. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 711. iT - 9.12e5T^{2} \)
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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

−14.20695022495807425616904333556, −12.79818346649065106231838430054, −11.00046033731787343899387677983, −9.993722001286644880528874868706, −9.040516892026406477815131149396, −8.629034337924555052467554854831, −7.19284280140987893944392887397, −6.42850669237812838602959620701, −4.21597189435462349857553714338, −1.76986841786520581488918630171, 0.57332089592461766302916744247, 2.60163641047063043982144673085, 3.34464888634603903369226454605, 6.72324547952454474727624760338, 7.65386052451512674367138176370, 8.776420835123757949928029287199, 9.460811960458410350171618733243, 10.45619678401992677355165968192, 11.59821920027834436818252771252, 12.65109025017533520884210686513

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