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Properties

Label 2-105-105.17-c2-0-10
Degree $2$
Conductor $105$
Sign $-0.262 + 0.964i$
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 − 0.875i)2-s + (−2.44 + 1.74i)3-s + (6.44 + 3.72i)4-s + (−4.71 + 1.65i)5-s + (9.50 − 3.55i)6-s + (1.33 + 6.87i)7-s + (−8.24 − 8.24i)8-s + (2.93 − 8.50i)9-s + (16.8 − 1.28i)10-s + (1.43 + 0.831i)11-s + (−22.2 + 2.13i)12-s + (−15.7 − 15.7i)13-s + (1.65 − 23.6i)14-s + (8.64 − 12.2i)15-s + (4.82 + 8.36i)16-s + (7.27 − 1.94i)17-s + ⋯
L(s)  = 1  + (−1.63 − 0.437i)2-s + (−0.814 + 0.580i)3-s + (1.61 + 0.930i)4-s + (−0.943 + 0.331i)5-s + (1.58 − 0.591i)6-s + (0.190 + 0.981i)7-s + (−1.03 − 1.03i)8-s + (0.326 − 0.945i)9-s + (1.68 − 0.128i)10-s + (0.130 + 0.0755i)11-s + (−1.85 + 0.177i)12-s + (−1.21 − 1.21i)13-s + (0.118 − 1.68i)14-s + (0.576 − 0.817i)15-s + (0.301 + 0.522i)16-s + (0.427 − 0.114i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.262 + 0.964i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.262 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $-0.262 + 0.964i$
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: \((2,\ 105,\ (\ :1),\ -0.262 + 0.964i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.102511 - 0.134091i\)
\(L(\frac12)\) \(\approx\) \(0.102511 - 0.134091i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.44 - 1.74i)T \)
5 \( 1 + (4.71 - 1.65i)T \)
7 \( 1 + (-1.33 - 6.87i)T \)
good2 \( 1 + (3.26 + 0.875i)T + (3.46 + 2i)T^{2} \)
11 \( 1 + (-1.43 - 0.831i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (15.7 + 15.7i)T + 169iT^{2} \)
17 \( 1 + (-7.27 + 1.94i)T + (250. - 144.5i)T^{2} \)
19 \( 1 + (6.28 + 10.8i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (-5.15 + 19.2i)T + (-458. - 264.5i)T^{2} \)
29 \( 1 - 18.4T + 841T^{2} \)
31 \( 1 + (-14.8 - 8.54i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-4.39 + 16.4i)T + (-1.18e3 - 684.5i)T^{2} \)
41 \( 1 + 57.3T + 1.68e3T^{2} \)
43 \( 1 + (-9.66 + 9.66i)T - 1.84e3iT^{2} \)
47 \( 1 + (-23.1 + 86.5i)T + (-1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (18.7 - 5.02i)T + (2.43e3 - 1.40e3i)T^{2} \)
59 \( 1 + (65.3 + 37.7i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (44.4 - 25.6i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (77.0 - 20.6i)T + (3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 - 34.3iT - 5.04e3T^{2} \)
73 \( 1 + (-47.3 + 12.6i)T + (4.61e3 - 2.66e3i)T^{2} \)
79 \( 1 + (-109. + 63.4i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-44.3 + 44.3i)T - 6.88e3iT^{2} \)
89 \( 1 + (61.8 - 35.7i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (73.5 - 73.5i)T - 9.40e3iT^{2} \)
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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

−12.26737020941017221928911904564, −11.98185456558189321174028636816, −10.81322578237485022787034845554, −10.16342438551873993755536239454, −8.997429135342959466175770033900, −7.980089957552542590447439740785, −6.78417809314214128036443729893, −5.00002591958511138875415081394, −2.87462723993017048155089874025, −0.25642112313697984064966578588, 1.32304459122089268466521545305, 4.58512942855600930316514313153, 6.52651047566714881826635757576, 7.42819367888700125735133780076, 8.031708929034490276551637883426, 9.495374724097790606604594467517, 10.57539614118360655547284139181, 11.48933346021209639819190136571, 12.33065151214277891617986999748, 13.90470236760424187071546800309

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