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Properties

Label 2-105-35.19-c2-0-10
Degree $2$
Conductor $105$
Sign $0.617 - 0.786i$
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  + (3.26 + 1.88i)2-s + (−0.866 − 1.5i)3-s + (5.12 + 8.86i)4-s + (0.907 − 4.91i)5-s − 6.53i·6-s + (1.16 + 6.90i)7-s + 23.5i·8-s + (−1.5 + 2.59i)9-s + (12.2 − 14.3i)10-s + (−8.95 − 15.5i)11-s + (8.86 − 15.3i)12-s + 1.69·13-s + (−9.21 + 24.7i)14-s + (−8.16 + 2.89i)15-s + (−23.9 + 41.5i)16-s + (−1.48 − 2.57i)17-s + ⋯
L(s)  = 1  + (1.63 + 0.943i)2-s + (−0.288 − 0.5i)3-s + (1.28 + 2.21i)4-s + (0.181 − 0.983i)5-s − 1.08i·6-s + (0.166 + 0.986i)7-s + 2.94i·8-s + (−0.166 + 0.288i)9-s + (1.22 − 1.43i)10-s + (−0.814 − 1.41i)11-s + (0.739 − 1.28i)12-s + 0.130·13-s + (−0.658 + 1.76i)14-s + (−0.544 + 0.193i)15-s + (−1.49 + 2.59i)16-s + (−0.0876 − 0.151i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.617 - 0.786i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.617 - 0.786i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.617 - 0.786i$
Analytic conductor: \(2.86104\)
Root analytic conductor: \(1.69146\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :1),\ 0.617 - 0.786i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.54467 + 1.23736i\)
\(L(\frac12)\) \(\approx\) \(2.54467 + 1.23736i\)
\(L(2)\) 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 \( 1 + (-0.907 + 4.91i)T \)
7 \( 1 + (-1.16 - 6.90i)T \)
good2 \( 1 + (-3.26 - 1.88i)T + (2 + 3.46i)T^{2} \)
11 \( 1 + (8.95 + 15.5i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 - 1.69T + 169T^{2} \)
17 \( 1 + (1.48 + 2.57i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (21.6 + 12.4i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-7.89 - 4.55i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 - 17.4T + 841T^{2} \)
31 \( 1 + (4.27 - 2.47i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-47.3 - 27.3i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 29.2iT - 1.68e3T^{2} \)
43 \( 1 - 18.1iT - 1.84e3T^{2} \)
47 \( 1 + (2.28 - 3.95i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-21.8 + 12.6i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (19.6 - 11.3i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-52.1 - 30.1i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (39.4 - 22.7i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 21.4T + 5.04e3T^{2} \)
73 \( 1 + (46.4 + 80.4i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (35.5 - 61.6i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 10.3T + 6.88e3T^{2} \)
89 \( 1 + (-10.8 - 6.27i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 127.T + 9.40e3T^{2} \)
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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

−13.30224598629537283980141888042, −13.13095479805889443560289067940, −12.02926276969553666971889044326, −11.19877645412004454377260737007, −8.699459085176683592506075850539, −7.989743256053644425396249212571, −6.35720309925782825431271666914, −5.60825747420915978483159997071, −4.67333408099953998231604853178, −2.74203258525676470082775980364, 2.26182899315679852659262642230, 3.80537417374635336406177058055, 4.74150879969335337981666730287, 6.13541862366255672541132470082, 7.26109055220027431506686334469, 10.01205761302940298667197653531, 10.47219065657703653538081341317, 11.22271992242439728434546508180, 12.46039259691862860564412946636, 13.28672086757203903713675799446

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