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Properties

Label 2-105-105.17-c2-0-11
Degree $2$
Conductor $105$
Sign $0.125 - 0.992i$
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  + (1.40 + 0.375i)2-s + (0.0184 + 2.99i)3-s + (−1.64 − 0.947i)4-s + (3.73 + 3.32i)5-s + (−1.10 + 4.21i)6-s + (2.97 + 6.33i)7-s + (−6.04 − 6.04i)8-s + (−8.99 + 0.110i)9-s + (3.98 + 6.06i)10-s + (3.76 + 2.17i)11-s + (2.81 − 4.94i)12-s + (4.76 + 4.76i)13-s + (1.78 + 9.99i)14-s + (−9.90 + 11.2i)15-s + (−2.41 − 4.18i)16-s + (19.0 − 5.10i)17-s + ⋯
L(s)  = 1  + (0.700 + 0.187i)2-s + (0.00614 + 0.999i)3-s + (−0.410 − 0.236i)4-s + (0.746 + 0.664i)5-s + (−0.183 + 0.701i)6-s + (0.424 + 0.905i)7-s + (−0.755 − 0.755i)8-s + (−0.999 + 0.0122i)9-s + (0.398 + 0.606i)10-s + (0.342 + 0.197i)11-s + (0.234 − 0.411i)12-s + (0.366 + 0.366i)13-s + (0.127 + 0.714i)14-s + (−0.660 + 0.751i)15-s + (−0.150 − 0.261i)16-s + (1.12 − 0.300i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.125 - 0.992i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.125 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.125 - 0.992i$
Analytic conductor: \(2.86104\)
Root analytic conductor: \(1.69146\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :1),\ 0.125 - 0.992i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.37244 + 1.20976i\)
\(L(\frac12)\) \(\approx\) \(1.37244 + 1.20976i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.0184 - 2.99i)T \)
5 \( 1 + (-3.73 - 3.32i)T \)
7 \( 1 + (-2.97 - 6.33i)T \)
good2 \( 1 + (-1.40 - 0.375i)T + (3.46 + 2i)T^{2} \)
11 \( 1 + (-3.76 - 2.17i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-4.76 - 4.76i)T + 169iT^{2} \)
17 \( 1 + (-19.0 + 5.10i)T + (250. - 144.5i)T^{2} \)
19 \( 1 + (12.5 + 21.8i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (-8.21 + 30.6i)T + (-458. - 264.5i)T^{2} \)
29 \( 1 - 28.8T + 841T^{2} \)
31 \( 1 + (12.3 + 7.12i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-13.1 + 49.0i)T + (-1.18e3 - 684.5i)T^{2} \)
41 \( 1 + 1.56T + 1.68e3T^{2} \)
43 \( 1 + (46.2 - 46.2i)T - 1.84e3iT^{2} \)
47 \( 1 + (-3.56 + 13.2i)T + (-1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (33.7 - 9.04i)T + (2.43e3 - 1.40e3i)T^{2} \)
59 \( 1 + (-54.4 - 31.4i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-9.31 + 5.37i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (112. - 30.0i)T + (3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 - 8.44iT - 5.04e3T^{2} \)
73 \( 1 + (-26.0 + 6.96i)T + (4.61e3 - 2.66e3i)T^{2} \)
79 \( 1 + (59.9 - 34.6i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-11.5 + 11.5i)T - 6.88e3iT^{2} \)
89 \( 1 + (58.5 - 33.8i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (-124. + 124. i)T - 9.40e3iT^{2} \)
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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

−14.20642778006380752849557759484, −12.92067121646396810104898973205, −11.64538617926009513824863867777, −10.50103353296838391432942713591, −9.472665974992287203586867250277, −8.708580460472303287588462671377, −6.49030288340733350466957005409, −5.51325064641215449276764511934, −4.47754773252886991001469819643, −2.85893842526836905884069377506, 1.38232636496825546879141076194, 3.50981856665234649246875856933, 5.13063752065841484172970111903, 6.17983699932091404795562582217, 7.85109034894976663332110828479, 8.654382754957340324556100553636, 10.13707685366689542464259745119, 11.64206193072520863975151188420, 12.51436675552982069205347317490, 13.34369433325505142631769070759

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