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Properties

Label 2-105-105.104-c3-0-20
Degree $2$
Conductor $105$
Sign $0.452 - 0.891i$
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  + 4.45·2-s + (−5.00 + 1.40i)3-s + 11.8·4-s + (0.892 + 11.1i)5-s + (−22.3 + 6.26i)6-s + (12.5 + 13.5i)7-s + 17.3·8-s + (23.0 − 14.0i)9-s + (3.98 + 49.7i)10-s + 30.2i·11-s + (−59.4 + 16.6i)12-s + 18.9·13-s + (56.1 + 60.5i)14-s + (−20.1 − 54.5i)15-s − 17.7·16-s + 4.59i·17-s + ⋯
L(s)  = 1  + 1.57·2-s + (−0.962 + 0.270i)3-s + 1.48·4-s + (0.0798 + 0.996i)5-s + (−1.51 + 0.426i)6-s + (0.680 + 0.733i)7-s + 0.766·8-s + (0.853 − 0.520i)9-s + (0.125 + 1.57i)10-s + 0.830i·11-s + (−1.43 + 0.401i)12-s + 0.404·13-s + (1.07 + 1.15i)14-s + (−0.346 − 0.938i)15-s − 0.277·16-s + 0.0655i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.452 - 0.891i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.452 - 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.452 - 0.891i$
Analytic conductor: \(6.19520\)
Root analytic conductor: \(2.48901\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (104, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :3/2),\ 0.452 - 0.891i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.42021 + 1.48623i\)
\(L(\frac12)\) \(\approx\) \(2.42021 + 1.48623i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (5.00 - 1.40i)T \)
5 \( 1 + (-0.892 - 11.1i)T \)
7 \( 1 + (-12.5 - 13.5i)T \)
good2 \( 1 - 4.45T + 8T^{2} \)
11 \( 1 - 30.2iT - 1.33e3T^{2} \)
13 \( 1 - 18.9T + 2.19e3T^{2} \)
17 \( 1 - 4.59iT - 4.91e3T^{2} \)
19 \( 1 + 119. iT - 6.85e3T^{2} \)
23 \( 1 - 134.T + 1.21e4T^{2} \)
29 \( 1 + 203. iT - 2.43e4T^{2} \)
31 \( 1 + 61.4iT - 2.97e4T^{2} \)
37 \( 1 + 337. iT - 5.06e4T^{2} \)
41 \( 1 - 135.T + 6.89e4T^{2} \)
43 \( 1 - 270. iT - 7.95e4T^{2} \)
47 \( 1 - 273. iT - 1.03e5T^{2} \)
53 \( 1 - 222.T + 1.48e5T^{2} \)
59 \( 1 - 735.T + 2.05e5T^{2} \)
61 \( 1 + 312. iT - 2.26e5T^{2} \)
67 \( 1 + 751. iT - 3.00e5T^{2} \)
71 \( 1 - 640. iT - 3.57e5T^{2} \)
73 \( 1 + 469.T + 3.89e5T^{2} \)
79 \( 1 - 126.T + 4.93e5T^{2} \)
83 \( 1 - 299. iT - 5.71e5T^{2} \)
89 \( 1 - 425.T + 7.04e5T^{2} \)
97 \( 1 + 561.T + 9.12e5T^{2} \)
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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

−13.37577480471093228428566231250, −12.45837735153445855549964168807, −11.37437705350490836352803071058, −11.02887749434363075597751607506, −9.429676169282431797440620688382, −7.24709756217392454281299657606, −6.25969834010269869074086658558, −5.24644850825990306243666645612, −4.19801643795680985657790428041, −2.50339549478326711357495140549, 1.27223839698767819623074496391, 3.83575925934601421534969815587, 4.98602121676742023846995088717, 5.71052022018252085550543844679, 7.00395663125751893334482444403, 8.490465199881179852213215351397, 10.44907505860028172482704374849, 11.44771184685219542689168463216, 12.21764482164604238703795571194, 13.17153363480259550154522319979

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