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Properties

Label 2-105-35.24-c2-0-14
Degree $2$
Conductor $105$
Sign $0.453 + 0.891i$
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  + (2.53 − 1.46i)2-s + (0.866 − 1.5i)3-s + (2.29 − 3.98i)4-s + (4.74 + 1.57i)5-s − 5.07i·6-s + (−6.93 + 0.976i)7-s − 1.75i·8-s + (−1.5 − 2.59i)9-s + (14.3 − 2.96i)10-s + (−2.31 + 4.00i)11-s + (−3.98 − 6.89i)12-s − 14.9·13-s + (−16.1 + 12.6i)14-s + (6.46 − 5.75i)15-s + (6.62 + 11.4i)16-s + (8.38 − 14.5i)17-s + ⋯
L(s)  = 1  + (1.26 − 0.733i)2-s + (0.288 − 0.5i)3-s + (0.574 − 0.995i)4-s + (0.949 + 0.314i)5-s − 0.846i·6-s + (−0.990 + 0.139i)7-s − 0.219i·8-s + (−0.166 − 0.288i)9-s + (1.43 − 0.296i)10-s + (−0.210 + 0.363i)11-s + (−0.331 − 0.574i)12-s − 1.15·13-s + (−1.15 + 0.903i)14-s + (0.431 − 0.383i)15-s + (0.414 + 0.717i)16-s + (0.493 − 0.853i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.453 + 0.891i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.453 + 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.453 + 0.891i$
Analytic conductor: \(2.86104\)
Root analytic conductor: \(1.69146\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (94, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :1),\ 0.453 + 0.891i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.29778 - 1.40933i\)
\(L(\frac12)\) \(\approx\) \(2.29778 - 1.40933i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.866 + 1.5i)T \)
5 \( 1 + (-4.74 - 1.57i)T \)
7 \( 1 + (6.93 - 0.976i)T \)
good2 \( 1 + (-2.53 + 1.46i)T + (2 - 3.46i)T^{2} \)
11 \( 1 + (2.31 - 4.00i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + 14.9T + 169T^{2} \)
17 \( 1 + (-8.38 + 14.5i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (-15.2 + 8.78i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (36.9 - 21.3i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 - 18.1T + 841T^{2} \)
31 \( 1 + (-3.46 - 2.00i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-24.5 + 14.1i)T + (684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 44.5iT - 1.68e3T^{2} \)
43 \( 1 - 44.3iT - 1.84e3T^{2} \)
47 \( 1 + (30.3 + 52.6i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (46.8 + 27.0i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-1.54 - 0.891i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (44.4 - 25.6i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-2.44 - 1.41i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 19.2T + 5.04e3T^{2} \)
73 \( 1 + (-49.4 + 85.6i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (59.7 + 103. i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 89.3T + 6.88e3T^{2} \)
89 \( 1 + (97.2 - 56.1i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 115.T + 9.40e3T^{2} \)
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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

−13.40129070441798206417161409370, −12.44249498405321649982883149896, −11.74697748783674444341625799219, −10.16499658762630156620903610887, −9.454160082148731235231010266038, −7.46766498342599159924238008086, −6.17648433633368256835582760844, −5.08553067934971793147258686329, −3.23035486594947934939317022316, −2.24109671236404261050652380941, 2.95083456992975228351838440436, 4.39329334873133819039049327253, 5.63817366194907213058057705441, 6.44977723178128046447034821456, 7.982862263459852510513086096273, 9.660782484901982784726701175106, 10.17015566660785837320516707180, 12.23769142755265684555309534695, 12.89940240555680836591844124325, 13.97886878831716693902715445352

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