Properties

Label 2-105-21.17-c3-0-9
Degree $2$
Conductor $105$
Sign $0.394 - 0.919i$
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  + (1.48 − 0.859i)2-s + (−4.11 − 3.16i)3-s + (−2.52 + 4.36i)4-s + (−2.5 − 4.33i)5-s + (−8.85 − 1.17i)6-s + (2.28 + 18.3i)7-s + 22.4i·8-s + (6.94 + 26.0i)9-s + (−7.44 − 4.29i)10-s + (49.7 + 28.7i)11-s + (24.2 − 10.0i)12-s − 44.6i·13-s + (19.2 + 25.4i)14-s + (−3.41 + 25.7i)15-s + (−0.881 − 1.52i)16-s + (−54.8 + 95.0i)17-s + ⋯
L(s)  = 1  + (0.526 − 0.304i)2-s + (−0.792 − 0.609i)3-s + (−0.315 + 0.545i)4-s + (−0.223 − 0.387i)5-s + (−0.602 − 0.0798i)6-s + (0.123 + 0.992i)7-s + 0.991i·8-s + (0.257 + 0.966i)9-s + (−0.235 − 0.135i)10-s + (1.36 + 0.787i)11-s + (0.582 − 0.240i)12-s − 0.952i·13-s + (0.366 + 0.485i)14-s + (−0.0587 + 0.443i)15-s + (−0.0137 − 0.0238i)16-s + (−0.783 + 1.35i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.394 - 0.919i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.394 - 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.394 - 0.919i$
Analytic conductor: \(6.19520\)
Root analytic conductor: \(2.48901\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :3/2),\ 0.394 - 0.919i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.963166 + 0.634994i\)
\(L(\frac12)\) \(\approx\) \(0.963166 + 0.634994i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (4.11 + 3.16i)T \)
5 \( 1 + (2.5 + 4.33i)T \)
7 \( 1 + (-2.28 - 18.3i)T \)
good2 \( 1 + (-1.48 + 0.859i)T + (4 - 6.92i)T^{2} \)
11 \( 1 + (-49.7 - 28.7i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 44.6iT - 2.19e3T^{2} \)
17 \( 1 + (54.8 - 95.0i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (79.5 - 45.9i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (27.1 - 15.6i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 183. iT - 2.43e4T^{2} \)
31 \( 1 + (-123. - 71.2i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (114. + 199. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 185.T + 6.89e4T^{2} \)
43 \( 1 + 22.3T + 7.95e4T^{2} \)
47 \( 1 + (116. + 201. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-400. - 230. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-229. + 397. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (248. - 143. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-416. + 720. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 471. iT - 3.57e5T^{2} \)
73 \( 1 + (-469. - 270. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (175. + 304. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 866.T + 5.71e5T^{2} \)
89 \( 1 + (-519. - 899. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 323. iT - 9.12e5T^{2} \)
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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

−12.90263895766722843548369138819, −12.49877502334576322216986055070, −11.83801083039424095823518185349, −10.68228265009911950201883875709, −8.907953449950020657323871281399, −8.040761934317482655397429381533, −6.48372788770320770714185867796, −5.25766258936561345273329370372, −4.02416101198493797787879080531, −1.94177179033482876521612735826, 0.62556341331199104973002430029, 3.97166299699308696989679954574, 4.61542095342670073024749159459, 6.30910737658951056858708721878, 6.83887523346271990029981170964, 9.027342341953038973084671073102, 9.964424737790061477639734421470, 11.11429152847817663241571834485, 11.74660805086640092678625560274, 13.45456667631054290167708178646

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