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Properties

Label 2-105-21.5-c3-0-4
Degree $2$
Conductor $105$
Sign $0.962 - 0.270i$
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  + (−3.06 − 1.76i)2-s + (−0.433 − 5.17i)3-s + (2.25 + 3.91i)4-s + (−2.5 + 4.33i)5-s + (−7.83 + 16.6i)6-s + (5.77 + 17.5i)7-s + 12.3i·8-s + (−26.6 + 4.48i)9-s + (15.3 − 8.84i)10-s + (3.05 − 1.76i)11-s + (19.2 − 13.3i)12-s + 14.5i·13-s + (13.4 − 64.1i)14-s + (23.5 + 11.0i)15-s + (39.8 − 69.0i)16-s + (30.9 + 53.5i)17-s + ⋯
L(s)  = 1  + (−1.08 − 0.625i)2-s + (−0.0834 − 0.996i)3-s + (0.282 + 0.488i)4-s + (−0.223 + 0.387i)5-s + (−0.532 + 1.13i)6-s + (0.311 + 0.950i)7-s + 0.544i·8-s + (−0.986 + 0.166i)9-s + (0.484 − 0.279i)10-s + (0.0837 − 0.0483i)11-s + (0.463 − 0.322i)12-s + 0.310i·13-s + (0.256 − 1.22i)14-s + (0.404 + 0.190i)15-s + (0.622 − 1.07i)16-s + (0.441 + 0.764i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 - 0.270i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.962 - 0.270i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.962 - 0.270i$
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: \((2,\ 105,\ (\ :3/2),\ 0.962 - 0.270i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.596183 + 0.0822087i\)
\(L(\frac12)\) \(\approx\) \(0.596183 + 0.0822087i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.433 + 5.17i)T \)
5 \( 1 + (2.5 - 4.33i)T \)
7 \( 1 + (-5.77 - 17.5i)T \)
good2 \( 1 + (3.06 + 1.76i)T + (4 + 6.92i)T^{2} \)
11 \( 1 + (-3.05 + 1.76i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 14.5iT - 2.19e3T^{2} \)
17 \( 1 + (-30.9 - 53.5i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (39.8 + 23.0i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-99.5 - 57.4i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 127. iT - 2.43e4T^{2} \)
31 \( 1 + (-183. + 106. i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (142. - 246. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 328.T + 6.89e4T^{2} \)
43 \( 1 + 108.T + 7.95e4T^{2} \)
47 \( 1 + (194. - 337. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-514. + 297. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-275. - 477. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-411. - 237. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-208. - 360. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 399. iT - 3.57e5T^{2} \)
73 \( 1 + (134. - 77.6i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (586. - 1.01e3i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 4.43T + 5.71e5T^{2} \)
89 \( 1 + (-505. + 875. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 27.5iT - 9.12e5T^{2} \)
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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

−13.06462939046041222529104905222, −11.85896679172455172988642297532, −11.37836015053865249231347160566, −10.16249554100606535684471373891, −8.782184826427943039425964181689, −8.190399052199526743159949475872, −6.81829966235382295330198779125, −5.41693250778521130903373122604, −2.77141482944528489159490805000, −1.44823708083455308876326826069, 0.52393673053980450353852327237, 3.70461820595981681128145097452, 5.00365030904741602235808866511, 6.73649815143774817375187271390, 7.976990144825120294593716916248, 8.824321293416444425240760551841, 9.932899507096613285368591101963, 10.62353349109928013999238007451, 11.93107764482088554515267563282, 13.39370501636024447306518821537

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