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Properties

Label 2-105-35.34-c2-0-6
Degree $2$
Conductor $105$
Sign $0.753 - 0.657i$
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  + 0.702i·2-s + 1.73·3-s + 3.50·4-s + (0.979 + 4.90i)5-s + 1.21i·6-s + (−3.48 − 6.07i)7-s + 5.27i·8-s + 2.99·9-s + (−3.44 + 0.687i)10-s + 5.39·11-s + 6.07·12-s − 12.5·13-s + (4.26 − 2.44i)14-s + (1.69 + 8.49i)15-s + 10.3·16-s + 8.14·17-s + ⋯
L(s)  = 1  + 0.351i·2-s + 0.577·3-s + 0.876·4-s + (0.195 + 0.980i)5-s + 0.202i·6-s + (−0.497 − 0.867i)7-s + 0.659i·8-s + 0.333·9-s + (−0.344 + 0.0687i)10-s + 0.490·11-s + 0.506·12-s − 0.963·13-s + (0.304 − 0.174i)14-s + (0.113 + 0.566i)15-s + 0.645·16-s + 0.479·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.753 - 0.657i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.753 - 0.657i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.753 - 0.657i$
Analytic conductor: \(2.86104\)
Root analytic conductor: \(1.69146\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (34, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :1),\ 0.753 - 0.657i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.73607 + 0.650997i\)
\(L(\frac12)\) \(\approx\) \(1.73607 + 0.650997i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 1.73T \)
5 \( 1 + (-0.979 - 4.90i)T \)
7 \( 1 + (3.48 + 6.07i)T \)
good2 \( 1 - 0.702iT - 4T^{2} \)
11 \( 1 - 5.39T + 121T^{2} \)
13 \( 1 + 12.5T + 169T^{2} \)
17 \( 1 - 8.14T + 289T^{2} \)
19 \( 1 - 1.94iT - 361T^{2} \)
23 \( 1 + 11.4iT - 529T^{2} \)
29 \( 1 + 44.2T + 841T^{2} \)
31 \( 1 + 50.4iT - 961T^{2} \)
37 \( 1 + 41.7iT - 1.36e3T^{2} \)
41 \( 1 + 36.0iT - 1.68e3T^{2} \)
43 \( 1 + 1.13iT - 1.84e3T^{2} \)
47 \( 1 + 24.3T + 2.20e3T^{2} \)
53 \( 1 - 88.3iT - 2.80e3T^{2} \)
59 \( 1 - 46.9iT - 3.48e3T^{2} \)
61 \( 1 + 47.5iT - 3.72e3T^{2} \)
67 \( 1 - 104. iT - 4.48e3T^{2} \)
71 \( 1 - 52.7T + 5.04e3T^{2} \)
73 \( 1 - 74.2T + 5.32e3T^{2} \)
79 \( 1 - 76.3T + 6.24e3T^{2} \)
83 \( 1 + 140.T + 6.88e3T^{2} \)
89 \( 1 - 33.4iT - 7.92e3T^{2} \)
97 \( 1 - 120.T + 9.40e3T^{2} \)
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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

−14.00305159038615778379550695934, −12.68716453595940269137081974200, −11.39371704511604642479212801865, −10.41677952989799775886949545355, −9.497552836648034005707806136015, −7.63909631124206317474823752570, −7.13096287455525719464748515852, −5.96038336172308215837217779821, −3.73901079484318952876061397590, −2.35208135161059601114750343095, 1.81853271019007613798239478909, 3.31159862053199815452361686344, 5.22388333799596880598260640791, 6.63753223490865835287519464788, 7.999023294353073531810197999228, 9.260920929484746994110569090539, 9.948254157884530883497904833023, 11.58540465210945625837012539813, 12.36412893378591216796496084041, 13.09213286550846050536945073851

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