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Properties

Label 2-105-35.9-c3-0-7
Degree $2$
Conductor $105$
Sign $0.320 + 0.947i$
Analytic cond. $6.19520$
Root an. cond. $2.48901$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.89 − 1.67i)2-s + (−2.59 + 1.5i)3-s + (1.58 + 2.73i)4-s + (9.21 − 6.33i)5-s + 10.0·6-s + (−6.15 + 17.4i)7-s + 16.1i·8-s + (4.5 − 7.79i)9-s + (−37.2 + 2.93i)10-s + (17.8 + 30.8i)11-s + (−8.21 − 4.74i)12-s − 88.7i·13-s + (46.9 − 40.2i)14-s + (−14.4 + 30.2i)15-s + (39.6 − 68.6i)16-s + (84.5 − 48.8i)17-s + ⋯
L(s)  = 1  + (−1.02 − 0.590i)2-s + (−0.499 + 0.288i)3-s + (0.197 + 0.342i)4-s + (0.824 − 0.566i)5-s + 0.681·6-s + (−0.332 + 0.943i)7-s + 0.714i·8-s + (0.166 − 0.288i)9-s + (−1.17 + 0.0928i)10-s + (0.488 + 0.845i)11-s + (−0.197 − 0.114i)12-s − 1.89i·13-s + (0.897 − 0.768i)14-s + (−0.248 + 0.521i)15-s + (0.619 − 1.07i)16-s + (1.20 − 0.696i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.320 + 0.947i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.320 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.320 + 0.947i$
Analytic conductor: \(6.19520\)
Root analytic conductor: \(2.48901\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :3/2),\ 0.320 + 0.947i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.668362 - 0.479193i\)
\(L(\frac12)\) \(\approx\) \(0.668362 - 0.479193i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.59 - 1.5i)T \)
5 \( 1 + (-9.21 + 6.33i)T \)
7 \( 1 + (6.15 - 17.4i)T \)
good2 \( 1 + (2.89 + 1.67i)T + (4 + 6.92i)T^{2} \)
11 \( 1 + (-17.8 - 30.8i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 88.7iT - 2.19e3T^{2} \)
17 \( 1 + (-84.5 + 48.8i)T + (2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-11.6 + 20.0i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (92.0 + 53.1i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 166.T + 2.43e4T^{2} \)
31 \( 1 + (20.1 + 34.8i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-42.1 - 24.3i)T + (2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 336.T + 6.89e4T^{2} \)
43 \( 1 + 268. iT - 7.95e4T^{2} \)
47 \( 1 + (-363. - 209. i)T + (5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (51.3 - 29.6i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (184. + 319. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (465. - 806. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-257. + 148. i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 715.T + 3.57e5T^{2} \)
73 \( 1 + (-709. + 409. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (108. - 188. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 194. iT - 5.71e5T^{2} \)
89 \( 1 + (-299. + 517. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 1.49e3iT - 9.12e5T^{2} \)
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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

−12.53601591476653359397870322021, −12.07977306503604408551826289310, −10.54632545301683571797942214361, −9.849677225336663697191769021455, −9.146180844326100126507563995652, −7.930011391150637470470219166067, −5.93278305802677547128444601981, −5.10988974897514466717035858744, −2.59289881860068972669569406656, −0.832432960524351997522181439733, 1.26558446549165476915760114805, 3.84427741371162409620059811703, 6.10892414120923541087783797381, 6.74216704797665503696101246955, 7.84021718280974915846154415591, 9.270112159623381174684097143107, 10.05020604073802201405735167383, 11.07922360126834088007007709749, 12.40930218281116738370055910216, 13.78248231331023891264777281112

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