Properties

Label 2-2106-13.12-c1-0-14
Degree $2$
Conductor $2106$
Sign $0.933 - 0.357i$
Analytic cond. $16.8164$
Root an. cond. $4.10079$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − i·2-s − 4-s + 0.594i·5-s + 1.67i·7-s + i·8-s + 0.594·10-s + 0.480i·11-s + (−1.28 − 3.36i)13-s + 1.67·14-s + 16-s − 2.09·17-s + 0.480i·19-s − 0.594i·20-s + 0.480·22-s + 3.66·23-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s + 0.265i·5-s + 0.633i·7-s + 0.353i·8-s + 0.188·10-s + 0.144i·11-s + (−0.357 − 0.933i)13-s + 0.448·14-s + 0.250·16-s − 0.507·17-s + 0.110i·19-s − 0.132i·20-s + 0.102·22-s + 0.764·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2106 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.933 - 0.357i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2106 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.933 - 0.357i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2106\)    =    \(2 \cdot 3^{4} \cdot 13\)
Sign: $0.933 - 0.357i$
Analytic conductor: \(16.8164\)
Root analytic conductor: \(4.10079\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2106} (649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2106,\ (\ :1/2),\ 0.933 - 0.357i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.390267187\)
\(L(\frac12)\) \(\approx\) \(1.390267187\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 \)
13 \( 1 + (1.28 + 3.36i)T \)
good5 \( 1 - 0.594iT - 5T^{2} \)
7 \( 1 - 1.67iT - 7T^{2} \)
11 \( 1 - 0.480iT - 11T^{2} \)
17 \( 1 + 2.09T + 17T^{2} \)
19 \( 1 - 0.480iT - 19T^{2} \)
23 \( 1 - 3.66T + 23T^{2} \)
29 \( 1 + 2.46T + 29T^{2} \)
31 \( 1 + 1.14iT - 31T^{2} \)
37 \( 1 - 3.65iT - 37T^{2} \)
41 \( 1 - 9.91iT - 41T^{2} \)
43 \( 1 - 6.91T + 43T^{2} \)
47 \( 1 - 6.24iT - 47T^{2} \)
53 \( 1 + 5.08T + 53T^{2} \)
59 \( 1 - 9.38iT - 59T^{2} \)
61 \( 1 - 7.81T + 61T^{2} \)
67 \( 1 - 14.3iT - 67T^{2} \)
71 \( 1 + 6.51iT - 71T^{2} \)
73 \( 1 - 5.91iT - 73T^{2} \)
79 \( 1 + 2.05T + 79T^{2} \)
83 \( 1 - 11.0iT - 83T^{2} \)
89 \( 1 - 9.48iT - 89T^{2} \)
97 \( 1 + 9.71iT - 97T^{2} \)
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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

−9.231597980883465496264358095399, −8.526601175024915804945052398636, −7.72246036721315884544414588157, −6.81941911133803314602986579794, −5.83584135194432876139242984646, −5.07744597472617940997560675198, −4.20475010863849624722591100561, −3.00308854412559457048486672721, −2.50833495644492011272492307069, −1.10403223827524402881554291442, 0.57368259809689300340975462514, 2.03651917406574040361386783000, 3.44968036387943290492321185710, 4.35713130298684675519749362565, 5.01952943904704463252018235157, 5.95449894863874984199085691235, 7.01347028194509932995601816113, 7.17504815612094746188127886536, 8.307705163376622722576304659500, 9.015156829508649543250523612702

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