Properties

Label 2-105-15.14-c2-0-17
Degree $2$
Conductor $105$
Sign $0.939 + 0.343i$
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  + 1.87·2-s + (2.98 − 0.302i)3-s − 0.494·4-s + (2.18 − 4.49i)5-s + (5.58 − 0.566i)6-s − 2.64i·7-s − 8.41·8-s + (8.81 − 1.80i)9-s + (4.08 − 8.42i)10-s + 20.6i·11-s + (−1.47 + 0.149i)12-s + 8.03i·13-s − 4.95i·14-s + (5.14 − 14.0i)15-s − 13.7·16-s − 18.0·17-s + ⋯
L(s)  = 1  + 0.936·2-s + (0.994 − 0.100i)3-s − 0.123·4-s + (0.436 − 0.899i)5-s + (0.931 − 0.0944i)6-s − 0.377i·7-s − 1.05·8-s + (0.979 − 0.200i)9-s + (0.408 − 0.842i)10-s + 1.88i·11-s + (−0.122 + 0.0124i)12-s + 0.618i·13-s − 0.353i·14-s + (0.343 − 0.939i)15-s − 0.861·16-s − 1.06·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.343i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.939 + 0.343i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.939 + 0.343i$
Analytic conductor: \(2.86104\)
Root analytic conductor: \(1.69146\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :1),\ 0.939 + 0.343i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.48243 - 0.439313i\)
\(L(\frac12)\) \(\approx\) \(2.48243 - 0.439313i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.98 + 0.302i)T \)
5 \( 1 + (-2.18 + 4.49i)T \)
7 \( 1 + 2.64iT \)
good2 \( 1 - 1.87T + 4T^{2} \)
11 \( 1 - 20.6iT - 121T^{2} \)
13 \( 1 - 8.03iT - 169T^{2} \)
17 \( 1 + 18.0T + 289T^{2} \)
19 \( 1 - 13.9T + 361T^{2} \)
23 \( 1 + 18.8T + 529T^{2} \)
29 \( 1 + 28.0iT - 841T^{2} \)
31 \( 1 - 14.1T + 961T^{2} \)
37 \( 1 + 46.9iT - 1.36e3T^{2} \)
41 \( 1 - 66.4iT - 1.68e3T^{2} \)
43 \( 1 - 2.30iT - 1.84e3T^{2} \)
47 \( 1 + 41.8T + 2.20e3T^{2} \)
53 \( 1 + 13.0T + 2.80e3T^{2} \)
59 \( 1 + 24.5iT - 3.48e3T^{2} \)
61 \( 1 - 68.2T + 3.72e3T^{2} \)
67 \( 1 + 72.3iT - 4.48e3T^{2} \)
71 \( 1 - 36.5iT - 5.04e3T^{2} \)
73 \( 1 + 42.4iT - 5.32e3T^{2} \)
79 \( 1 - 75.4T + 6.24e3T^{2} \)
83 \( 1 - 45.9T + 6.88e3T^{2} \)
89 \( 1 - 14.4iT - 7.92e3T^{2} \)
97 \( 1 - 132. iT - 9.40e3T^{2} \)
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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

−13.45576082430451680496751266678, −12.80402263132515476448862846660, −11.93332498037880645263220985248, −9.779303485420427010484346278544, −9.332073621829267883512322225119, −7.993089534934830740516640726060, −6.58701443667209943622287156746, −4.78904684678767792623266451353, −4.11982140597452207403898749536, −2.11057569115319628356813365199, 2.78061098455977943778945178278, 3.63314509176244296254398732821, 5.40518563695171351917322616404, 6.55175921677953304665892886107, 8.246667717544761752368267264958, 9.140279173190428002643763849670, 10.40218952757699703085648729267, 11.61953805439795415420531691780, 13.06610947113377368278582269646, 13.75514308218241357260956854565

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