Properties

Label 2-105-21.5-c3-0-30
Degree $2$
Conductor $105$
Sign $-0.993 - 0.117i$
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  + (−1.20 − 0.694i)2-s + (3.08 − 4.18i)3-s + (−3.03 − 5.25i)4-s + (−2.5 + 4.33i)5-s + (−6.60 + 2.88i)6-s + (−11.1 − 14.8i)7-s + 19.5i·8-s + (−7.99 − 25.7i)9-s + (6.01 − 3.47i)10-s + (−11.5 + 6.66i)11-s + (−31.3 − 3.50i)12-s + 27.3i·13-s + (3.09 + 25.5i)14-s + (10.4 + 23.8i)15-s + (−10.7 + 18.5i)16-s + (−8.19 − 14.2i)17-s + ⋯
L(s)  = 1  + (−0.425 − 0.245i)2-s + (0.593 − 0.805i)3-s + (−0.379 − 0.657i)4-s + (−0.223 + 0.387i)5-s + (−0.449 + 0.196i)6-s + (−0.600 − 0.799i)7-s + 0.863i·8-s + (−0.296 − 0.955i)9-s + (0.190 − 0.109i)10-s + (−0.316 + 0.182i)11-s + (−0.754 − 0.0844i)12-s + 0.583i·13-s + (0.0590 + 0.487i)14-s + (0.179 + 0.409i)15-s + (−0.167 + 0.290i)16-s + (−0.116 − 0.202i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 - 0.117i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.993 - 0.117i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $-0.993 - 0.117i$
Analytic conductor: \(6.19520\)
Root analytic conductor: \(2.48901\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :3/2),\ -0.993 - 0.117i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0413725 + 0.702404i\)
\(L(\frac12)\) \(\approx\) \(0.0413725 + 0.702404i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-3.08 + 4.18i)T \)
5 \( 1 + (2.5 - 4.33i)T \)
7 \( 1 + (11.1 + 14.8i)T \)
good2 \( 1 + (1.20 + 0.694i)T + (4 + 6.92i)T^{2} \)
11 \( 1 + (11.5 - 6.66i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 27.3iT - 2.19e3T^{2} \)
17 \( 1 + (8.19 + 14.2i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-3.26 - 1.88i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (171. + 98.9i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 68.2iT - 2.43e4T^{2} \)
31 \( 1 + (4.82 - 2.78i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-125. + 217. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 113.T + 6.89e4T^{2} \)
43 \( 1 - 15.9T + 7.95e4T^{2} \)
47 \( 1 + (-199. + 345. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-557. + 321. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (254. + 440. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-315. - 182. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (528. + 914. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 761. iT - 3.57e5T^{2} \)
73 \( 1 + (-399. + 230. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (392. - 680. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 1.28e3T + 5.71e5T^{2} \)
89 \( 1 + (-111. + 192. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 1.51e3iT - 9.12e5T^{2} \)
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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

−12.88840672657581404778885262301, −11.63945449630741238997542931132, −10.40153849539123250617872449315, −9.547588129232248002379150196358, −8.365326464153328890749911064161, −7.21293927853096512119792142753, −6.09144628906937390295129261093, −4.06117198296828457351379462389, −2.23246532749687742467679166979, −0.40949830193974173037980147791, 2.94008497633848623204907536680, 4.17268704574299827643352375878, 5.70537906749537296877409449194, 7.64818534199456759214184309719, 8.500667307815603982586852869084, 9.325849922005961734368813748174, 10.23585805164046305859603274616, 11.82718853215556876739583217949, 12.87338674396737534957455007109, 13.73729875155397438049572645031

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