Properties

Label 2-105-105.44-c2-0-17
Degree $2$
Conductor $105$
Sign $0.988 + 0.148i$
Analytic cond. $2.86104$
Root an. cond. $1.69146$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.949 − 1.64i)2-s + (2.65 + 1.40i)3-s + (0.196 + 0.340i)4-s + (0.399 + 4.98i)5-s + (4.82 − 3.02i)6-s + (−4.60 − 5.26i)7-s + 8.34·8-s + (5.04 + 7.45i)9-s + (8.57 + 4.07i)10-s + (−8.35 + 4.82i)11-s + (0.0423 + 1.17i)12-s − 16.9i·13-s + (−13.0 + 2.57i)14-s + (−5.94 + 13.7i)15-s + (7.13 − 12.3i)16-s + (−12.1 − 21.1i)17-s + ⋯
L(s)  = 1  + (0.474 − 0.822i)2-s + (0.883 + 0.468i)3-s + (0.0490 + 0.0850i)4-s + (0.0799 + 0.996i)5-s + (0.804 − 0.504i)6-s + (−0.658 − 0.752i)7-s + 1.04·8-s + (0.560 + 0.827i)9-s + (0.857 + 0.407i)10-s + (−0.759 + 0.438i)11-s + (0.00353 + 0.0981i)12-s − 1.30i·13-s + (−0.931 + 0.184i)14-s + (−0.396 + 0.918i)15-s + (0.446 − 0.772i)16-s + (−0.716 − 1.24i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.148i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.988 + 0.148i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.988 + 0.148i$
Analytic conductor: \(2.86104\)
Root analytic conductor: \(1.69146\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (44, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :1),\ 0.988 + 0.148i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.19468 - 0.163754i\)
\(L(\frac12)\) \(\approx\) \(2.19468 - 0.163754i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.65 - 1.40i)T \)
5 \( 1 + (-0.399 - 4.98i)T \)
7 \( 1 + (4.60 + 5.26i)T \)
good2 \( 1 + (-0.949 + 1.64i)T + (-2 - 3.46i)T^{2} \)
11 \( 1 + (8.35 - 4.82i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + 16.9iT - 169T^{2} \)
17 \( 1 + (12.1 + 21.1i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (-6.95 + 12.0i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (0.354 - 0.614i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 - 16.5iT - 841T^{2} \)
31 \( 1 + (-7.12 - 12.3i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (1.08 + 0.626i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 24.1iT - 1.68e3T^{2} \)
43 \( 1 - 57.6iT - 1.84e3T^{2} \)
47 \( 1 + (16.0 - 27.8i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-8.67 - 15.0i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (75.8 - 43.8i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-52.2 + 90.4i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-86.1 + 49.7i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 50.7iT - 5.04e3T^{2} \)
73 \( 1 + (81.5 - 47.0i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (3.71 - 6.43i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 69.6T + 6.88e3T^{2} \)
89 \( 1 + (-78.3 - 45.2i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 90.4iT - 9.40e3T^{2} \)
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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

−13.44816549454241828854165849297, −12.74195418452380776173270769230, −11.11747791796493526856962279146, −10.46066037249972862241507410491, −9.631079360291896521288544024019, −7.79941973066969503609195794003, −7.06026938854340389526027809473, −4.78278408134890098397866452442, −3.30834957273840650663355511119, −2.66125117462803068547134555250, 1.95696981453703377473036784161, 4.11946729931104834648632794084, 5.67648714672709117777227426704, 6.64362508531797206542521307318, 8.045864308849611478340429677565, 8.893574084359446267075408203816, 10.03787770802063919260472344353, 11.83775331206096983829080858055, 13.00350670535452213555197251783, 13.50421473769890785499942212903

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