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Properties

Label 2-105-105.17-c2-0-21
Degree $2$
Conductor $105$
Sign $0.982 + 0.186i$
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  + (2.63 + 0.704i)2-s + (−0.254 − 2.98i)3-s + (2.95 + 1.70i)4-s + (2.97 + 4.02i)5-s + (1.43 − 8.04i)6-s + (6.12 − 3.38i)7-s + (−1.12 − 1.12i)8-s + (−8.87 + 1.52i)9-s + (4.98 + 12.6i)10-s + (−5.63 − 3.25i)11-s + (4.35 − 9.27i)12-s + (13.0 + 13.0i)13-s + (18.5 − 4.58i)14-s + (11.2 − 9.90i)15-s + (−8.99 − 15.5i)16-s + (−30.7 + 8.22i)17-s + ⋯
L(s)  = 1  + (1.31 + 0.352i)2-s + (−0.0847 − 0.996i)3-s + (0.739 + 0.426i)4-s + (0.594 + 0.804i)5-s + (0.239 − 1.34i)6-s + (0.875 − 0.483i)7-s + (−0.140 − 0.140i)8-s + (−0.985 + 0.168i)9-s + (0.498 + 1.26i)10-s + (−0.512 − 0.295i)11-s + (0.362 − 0.772i)12-s + (1.00 + 1.00i)13-s + (1.32 − 0.327i)14-s + (0.750 − 0.660i)15-s + (−0.562 − 0.974i)16-s + (−1.80 + 0.484i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 + 0.186i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.982 + 0.186i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.982 + 0.186i$
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: \((2,\ 105,\ (\ :1),\ 0.982 + 0.186i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.56046 - 0.241250i\)
\(L(\frac12)\) \(\approx\) \(2.56046 - 0.241250i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.254 + 2.98i)T \)
5 \( 1 + (-2.97 - 4.02i)T \)
7 \( 1 + (-6.12 + 3.38i)T \)
good2 \( 1 + (-2.63 - 0.704i)T + (3.46 + 2i)T^{2} \)
11 \( 1 + (5.63 + 3.25i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-13.0 - 13.0i)T + 169iT^{2} \)
17 \( 1 + (30.7 - 8.22i)T + (250. - 144.5i)T^{2} \)
19 \( 1 + (-4.50 - 7.80i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (0.794 - 2.96i)T + (-458. - 264.5i)T^{2} \)
29 \( 1 + 21.7T + 841T^{2} \)
31 \( 1 + (15.7 + 9.07i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-5.32 + 19.8i)T + (-1.18e3 - 684.5i)T^{2} \)
41 \( 1 - 60.4T + 1.68e3T^{2} \)
43 \( 1 + (-16.3 + 16.3i)T - 1.84e3iT^{2} \)
47 \( 1 + (-9.02 + 33.6i)T + (-1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (15.7 - 4.22i)T + (2.43e3 - 1.40e3i)T^{2} \)
59 \( 1 + (-55.9 - 32.2i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-52.7 + 30.4i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (6.94 - 1.86i)T + (3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + 40.3iT - 5.04e3T^{2} \)
73 \( 1 + (85.5 - 22.9i)T + (4.61e3 - 2.66e3i)T^{2} \)
79 \( 1 + (-34.2 + 19.7i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-16.2 + 16.2i)T - 6.88e3iT^{2} \)
89 \( 1 + (5.70 - 3.29i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (56.3 - 56.3i)T - 9.40e3iT^{2} \)
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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

−13.57388694307804665394855104468, −12.98288777384017743478641774478, −11.47649746453431900747867357895, −10.91039704543646545932694804134, −8.925798535606505616648991983490, −7.41061037355901295886567529498, −6.47851843640738078917950980985, −5.59734648048290403688241763983, −3.97021443178218194609112326908, −2.13761496499117697400199729436, 2.53879071378789207676363121856, 4.28326036742749359697406221223, 5.11915920839712432423368653040, 5.91276825054319046284205507515, 8.403957422457395860219723948815, 9.223596544270847095227863055487, 10.80909050947498195313305407011, 11.44563428102205894089796613182, 12.75695506632844501203399234440, 13.45116451422740984741864608810

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