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Properties

Label 2-2106-9.4-c1-0-35
Degree $2$
Conductor $2106$
Sign $0.173 + 0.984i$
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  + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (−1 + 1.73i)5-s + (−2 − 3.46i)7-s − 0.999·8-s − 1.99·10-s + (2 + 3.46i)11-s + (−0.5 + 0.866i)13-s + (1.99 − 3.46i)14-s + (−0.5 − 0.866i)16-s + 2·17-s − 8·19-s + (−0.999 − 1.73i)20-s + (−1.99 + 3.46i)22-s + (0.500 + 0.866i)25-s − 0.999·26-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.447 + 0.774i)5-s + (−0.755 − 1.30i)7-s − 0.353·8-s − 0.632·10-s + (0.603 + 1.04i)11-s + (−0.138 + 0.240i)13-s + (0.534 − 0.925i)14-s + (−0.125 − 0.216i)16-s + 0.485·17-s − 1.83·19-s + (−0.223 − 0.387i)20-s + (−0.426 + 0.738i)22-s + (0.100 + 0.173i)25-s − 0.196·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2106 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2106 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2106\)    =    \(2 \cdot 3^{4} \cdot 13\)
Sign: $0.173 + 0.984i$
Analytic conductor: \(16.8164\)
Root analytic conductor: \(4.10079\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2106} (1405, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2106,\ (\ :1/2),\ 0.173 + 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4834786420\)
\(L(\frac12)\) \(\approx\) \(0.4834786420\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 \)
13 \( 1 + (0.5 - 0.866i)T \)
good5 \( 1 + (1 - 1.73i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (2 + 3.46i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2 - 3.46i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 - 2T + 17T^{2} \)
19 \( 1 + 8T + 19T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3 + 5.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2 + 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 + (-5 + 8.66i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2 + 3.46i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4 + 6.92i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 10T + 53T^{2} \)
59 \( 1 + (2 - 3.46i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1 - 1.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-8 + 13.8i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 - 2T + 73T^{2} \)
79 \( 1 + (4 + 6.92i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (6 + 10.3i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 14T + 89T^{2} \)
97 \( 1 + (5 + 8.66i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.882204707245729639133251202750, −7.77093856022445267993631519569, −7.25723312310500697443651956905, −6.67692746492188835259554080275, −6.08945291311261957472758902633, −4.67809671467951452037535282676, −4.01186201556154754661209449785, −3.44414002986291886611269865326, −2.07991402950745603907448599566, −0.15454200569578258202671511413, 1.30065482510390929304246838235, 2.63490972280806157145185330029, 3.37116307495136432418963639261, 4.35635858891677468611969633572, 5.20958086036343826360680953489, 6.06766350416311642568376554146, 6.56939892921063731835185650909, 8.125527618259877548524029964704, 8.616597230179061777533022413734, 9.198914538413569305673129950575

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