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Properties

Label 2-105-35.19-c2-0-6
Degree $2$
Conductor $105$
Sign $-0.975 + 0.221i$
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 + (−3.80 + 3.24i)5-s − 6.53i·6-s + (−1.16 − 6.90i)7-s − 23.5i·8-s + (−1.5 + 2.59i)9-s + (18.5 − 3.42i)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 + (−0.760 + 0.648i)5-s − 1.08i·6-s + (−0.166 − 0.986i)7-s − 2.94i·8-s + (−0.166 + 0.288i)9-s + (1.85 − 0.342i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.975 + 0.221i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.975 + 0.221i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $-0.975 + 0.221i$
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: \((2,\ 105,\ (\ :1),\ -0.975 + 0.221i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0231309 - 0.206672i\)
\(L(\frac12)\) \(\approx\) \(0.0231309 - 0.206672i\)
\(L(2)\) 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 \( 1 + (3.80 - 3.24i)T \)
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

−12.68836842975558622112244857099, −11.28992554084398718698967251078, −10.75399554573433051634646025793, −10.10784677550207193506559176971, −8.660917959315925554152520844033, −7.984047129803123362537339432975, −6.84024630176144177655295967843, −3.87280801295971834917823019678, −2.80100240870504257869227364934, −0.23815734608604652553675783759, 1.98609378418644466820561485000, 5.16700546701217115477719236158, 6.60026469626712640339583339808, 7.72466521442988107281924512970, 8.386909582518816140109300119807, 9.329906260715824157227468874358, 10.37588637282351711463691103595, 11.88183219969580402808853364589, 12.74079666931103590522691265037, 14.59431436536567170374765972215

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