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Properties

Label 2-105-35.24-c2-0-9
Degree $2$
Conductor $105$
Sign $0.347 + 0.937i$
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  + (−1.32 + 0.765i)2-s + (0.866 − 1.5i)3-s + (−0.829 + 1.43i)4-s + (−3.72 − 3.33i)5-s + 2.65i·6-s + (6.34 − 2.94i)7-s − 8.65i·8-s + (−1.5 − 2.59i)9-s + (7.48 + 1.56i)10-s + (6.68 − 11.5i)11-s + (1.43 + 2.48i)12-s − 15.5·13-s + (−6.15 + 8.76i)14-s + (−8.22 + 2.70i)15-s + (3.30 + 5.73i)16-s + (10.4 − 18.0i)17-s + ⋯
L(s)  = 1  + (−0.662 + 0.382i)2-s + (0.288 − 0.5i)3-s + (−0.207 + 0.358i)4-s + (−0.745 − 0.666i)5-s + 0.441i·6-s + (0.907 − 0.421i)7-s − 1.08i·8-s + (−0.166 − 0.288i)9-s + (0.748 + 0.156i)10-s + (0.607 − 1.05i)11-s + (0.119 + 0.207i)12-s − 1.19·13-s + (−0.439 + 0.626i)14-s + (−0.548 + 0.180i)15-s + (0.206 + 0.358i)16-s + (0.613 − 1.06i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.347 + 0.937i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.347 + 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.347 + 0.937i$
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: \((2,\ 105,\ (\ :1),\ 0.347 + 0.937i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.674439 - 0.469258i\)
\(L(\frac12)\) \(\approx\) \(0.674439 - 0.469258i\)
\(L(2)\) 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 \( 1 + (3.72 + 3.33i)T \)
7 \( 1 + (-6.34 + 2.94i)T \)
good2 \( 1 + (1.32 - 0.765i)T + (2 - 3.46i)T^{2} \)
11 \( 1 + (-6.68 + 11.5i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + 15.5T + 169T^{2} \)
17 \( 1 + (-10.4 + 18.0i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (-23.3 + 13.4i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (20.1 - 11.6i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + 43.9T + 841T^{2} \)
31 \( 1 + (-27.3 - 15.7i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (30.6 - 17.6i)T + (684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 19.4iT - 1.68e3T^{2} \)
43 \( 1 + 18.7iT - 1.84e3T^{2} \)
47 \( 1 + (-1.46 - 2.53i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-51.8 - 29.9i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-14.7 - 8.49i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-83.2 + 48.0i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (29.6 + 17.1i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 76.6T + 5.04e3T^{2} \)
73 \( 1 + (-23.9 + 41.4i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (25.8 + 44.7i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 13.1T + 6.88e3T^{2} \)
89 \( 1 + (46.8 - 27.0i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 95.2T + 9.40e3T^{2} \)
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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

−13.44552497643919042433007975512, −12.06671452703894917102316264240, −11.56660901550642899216684095903, −9.670005169385696942027219114484, −8.728767783112319402409217614502, −7.75538534581168443148122064122, −7.21490374749916563696752553893, −5.06846261334871435143191198891, −3.54745085485129228428549668614, −0.78514380399944799619125480170, 2.08876952995438800685260813154, 4.10616346017315941384923581806, 5.45417488796705252547834581857, 7.46750060716047664060562078920, 8.349575982044672728393904694027, 9.682519695498294272554272475366, 10.29785844264554325911999201061, 11.51585672896701280813525010788, 12.20316180230500607664023735662, 14.36750077782053698249148748754

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