Properties

Label 2-105-5.4-c3-0-15
Degree $2$
Conductor $105$
Sign $0.379 - 0.925i$
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  − 5.18i·2-s − 3i·3-s − 18.9·4-s + (−4.24 + 10.3i)5-s − 15.5·6-s − 7i·7-s + 56.5i·8-s − 9·9-s + (53.6 + 22.0i)10-s − 35.9·11-s + 56.7i·12-s + 45.2i·13-s − 36.3·14-s + (31.0 + 12.7i)15-s + 142.·16-s − 113. i·17-s + ⋯
L(s)  = 1  − 1.83i·2-s − 0.577i·3-s − 2.36·4-s + (−0.379 + 0.925i)5-s − 1.05·6-s − 0.377i·7-s + 2.49i·8-s − 0.333·9-s + (1.69 + 0.695i)10-s − 0.985·11-s + 1.36i·12-s + 0.965i·13-s − 0.693·14-s + (0.534 + 0.219i)15-s + 2.21·16-s − 1.61i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.379 - 0.925i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.379 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.379 - 0.925i$
Analytic conductor: \(6.19520\)
Root analytic conductor: \(2.48901\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :3/2),\ 0.379 - 0.925i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.207912 + 0.139437i\)
\(L(\frac12)\) \(\approx\) \(0.207912 + 0.139437i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 3iT \)
5 \( 1 + (4.24 - 10.3i)T \)
7 \( 1 + 7iT \)
good2 \( 1 + 5.18iT - 8T^{2} \)
11 \( 1 + 35.9T + 1.33e3T^{2} \)
13 \( 1 - 45.2iT - 2.19e3T^{2} \)
17 \( 1 + 113. iT - 4.91e3T^{2} \)
19 \( 1 + 61.5T + 6.85e3T^{2} \)
23 \( 1 + 30.6iT - 1.21e4T^{2} \)
29 \( 1 + 214.T + 2.43e4T^{2} \)
31 \( 1 - 164.T + 2.97e4T^{2} \)
37 \( 1 - 410. iT - 5.06e4T^{2} \)
41 \( 1 + 309.T + 6.89e4T^{2} \)
43 \( 1 + 29.9iT - 7.95e4T^{2} \)
47 \( 1 + 483. iT - 1.03e5T^{2} \)
53 \( 1 + 295. iT - 1.48e5T^{2} \)
59 \( 1 + 416.T + 2.05e5T^{2} \)
61 \( 1 + 151.T + 2.26e5T^{2} \)
67 \( 1 + 89.5iT - 3.00e5T^{2} \)
71 \( 1 - 714.T + 3.57e5T^{2} \)
73 \( 1 + 1.13e3iT - 3.89e5T^{2} \)
79 \( 1 - 323.T + 4.93e5T^{2} \)
83 \( 1 - 297. iT - 5.71e5T^{2} \)
89 \( 1 - 90.2T + 7.04e5T^{2} \)
97 \( 1 - 492. iT - 9.12e5T^{2} \)
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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

−12.08930858452417692122269517174, −11.44918798658959091150835948609, −10.58368085902245432827294698463, −9.620510626423757557516634153616, −8.214355049958738348785633582155, −6.87202622241605779877537475179, −4.77370533576422905793775935756, −3.28246318049781549774943209161, −2.14837348286011675999983431706, −0.13608906542958302754975368007, 4.02165309549407933467306641639, 5.22640840953907941668340536240, 5.97217843661931026807607552208, 7.76918477268921503445986724811, 8.351162358393976889089994031099, 9.335638264434699905763048422962, 10.65673932950181751339238282581, 12.61877681590840568800751756140, 13.19114425501968282548832035258, 14.61117206978981457499800122940

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