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Properties

Label 2-105-105.17-c2-0-13
Degree $2$
Conductor $105$
Sign $0.439 + 0.898i$
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  + (−2.58 − 0.691i)2-s + (1.70 − 2.46i)3-s + (2.71 + 1.56i)4-s + (4.71 + 1.64i)5-s + (−6.10 + 5.18i)6-s + (6.61 + 2.27i)7-s + (1.62 + 1.62i)8-s + (−3.18 − 8.41i)9-s + (−11.0 − 7.52i)10-s + (1.43 + 0.830i)11-s + (8.50 − 4.03i)12-s + (−3.87 − 3.87i)13-s + (−15.5 − 10.4i)14-s + (12.1 − 8.83i)15-s + (−9.35 − 16.1i)16-s + (22.4 − 6.02i)17-s + ⋯
L(s)  = 1  + (−1.29 − 0.345i)2-s + (0.568 − 0.822i)3-s + (0.679 + 0.392i)4-s + (0.943 + 0.329i)5-s + (−1.01 + 0.864i)6-s + (0.945 + 0.325i)7-s + (0.203 + 0.203i)8-s + (−0.353 − 0.935i)9-s + (−1.10 − 0.752i)10-s + (0.130 + 0.0754i)11-s + (0.709 − 0.335i)12-s + (−0.298 − 0.298i)13-s + (−1.10 − 0.747i)14-s + (0.808 − 0.588i)15-s + (−0.584 − 1.01i)16-s + (1.32 − 0.354i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.439 + 0.898i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.439 + 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.439 + 0.898i$
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: \((2,\ 105,\ (\ :1),\ 0.439 + 0.898i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.878905 - 0.548622i\)
\(L(\frac12)\) \(\approx\) \(0.878905 - 0.548622i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.70 + 2.46i)T \)
5 \( 1 + (-4.71 - 1.64i)T \)
7 \( 1 + (-6.61 - 2.27i)T \)
good2 \( 1 + (2.58 + 0.691i)T + (3.46 + 2i)T^{2} \)
11 \( 1 + (-1.43 - 0.830i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (3.87 + 3.87i)T + 169iT^{2} \)
17 \( 1 + (-22.4 + 6.02i)T + (250. - 144.5i)T^{2} \)
19 \( 1 + (11.4 + 19.8i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (2.50 - 9.35i)T + (-458. - 264.5i)T^{2} \)
29 \( 1 + 42.9T + 841T^{2} \)
31 \( 1 + (-15.6 - 9.00i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (14.8 - 55.3i)T + (-1.18e3 - 684.5i)T^{2} \)
41 \( 1 - 15.2T + 1.68e3T^{2} \)
43 \( 1 + (-16.7 + 16.7i)T - 1.84e3iT^{2} \)
47 \( 1 + (-11.0 + 41.3i)T + (-1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (24.7 - 6.61i)T + (2.43e3 - 1.40e3i)T^{2} \)
59 \( 1 + (-29.6 - 17.1i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (77.4 - 44.7i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-98.1 + 26.3i)T + (3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + 28.7iT - 5.04e3T^{2} \)
73 \( 1 + (7.99 - 2.14i)T + (4.61e3 - 2.66e3i)T^{2} \)
79 \( 1 + (81.2 - 46.9i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (83.3 - 83.3i)T - 6.88e3iT^{2} \)
89 \( 1 + (72.1 - 41.6i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (-55.5 + 55.5i)T - 9.40e3iT^{2} \)
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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

−13.44427926147230766448629351733, −12.08792086957791011430216580431, −11.04840282407567607708810722420, −9.863879583263291919508817904379, −9.017711947794586362185914800793, −8.034497721344270929567828036897, −7.05799428087489099082738619442, −5.40277650485311207372899022364, −2.59986326386983408290991948877, −1.39498800831982240362241282237, 1.76537548873152645571581456530, 4.19812830737016885498636888450, 5.71941109626510015031339329895, 7.58609638501685263543204502417, 8.410306661805443521283209267777, 9.387815269907675232668222683238, 10.13501306041024131074542719542, 10.97376148474753784248194639892, 12.75870634379205116683270729230, 14.15472994660767856880427977923

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