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Properties

Label 2-2106-13.12-c1-0-37
Degree $2$
Conductor $2106$
Sign $-0.316 + 0.948i$
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 − 0.484i·5-s − 5.05i·7-s + i·8-s − 0.484·10-s + 3.21i·11-s + (3.42 + 1.14i)13-s − 5.05·14-s + 16-s + 4.20·17-s + 3.21i·19-s + 0.484i·20-s + 3.21·22-s + 6.27·23-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s − 0.216i·5-s − 1.91i·7-s + 0.353i·8-s − 0.153·10-s + 0.968i·11-s + (0.948 + 0.316i)13-s − 1.35·14-s + 0.250·16-s + 1.01·17-s + 0.736i·19-s + 0.108i·20-s + 0.684·22-s + 1.30·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2106 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.316 + 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2106 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.316 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2106\)    =    \(2 \cdot 3^{4} \cdot 13\)
Sign: $-0.316 + 0.948i$
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: \((2,\ 2106,\ (\ :1/2),\ -0.316 + 0.948i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.831071507\)
\(L(\frac12)\) \(\approx\) \(1.831071507\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.42 - 1.14i)T \)
good5 \( 1 + 0.484iT - 5T^{2} \)
7 \( 1 + 5.05iT - 7T^{2} \)
11 \( 1 - 3.21iT - 11T^{2} \)
17 \( 1 - 4.20T + 17T^{2} \)
19 \( 1 - 3.21iT - 19T^{2} \)
23 \( 1 - 6.27T + 23T^{2} \)
29 \( 1 - 4.59T + 29T^{2} \)
31 \( 1 + 6.48iT - 31T^{2} \)
37 \( 1 + 2.08iT - 37T^{2} \)
41 \( 1 + 11.0iT - 41T^{2} \)
43 \( 1 + 9.46T + 43T^{2} \)
47 \( 1 - 5.28iT - 47T^{2} \)
53 \( 1 - 6.41T + 53T^{2} \)
59 \( 1 - 3.62iT - 59T^{2} \)
61 \( 1 + 1.00T + 61T^{2} \)
67 \( 1 - 1.08iT - 67T^{2} \)
71 \( 1 - 4.63iT - 71T^{2} \)
73 \( 1 + 0.325iT - 73T^{2} \)
79 \( 1 + 7.83T + 79T^{2} \)
83 \( 1 - 5.86iT - 83T^{2} \)
89 \( 1 + 8.42iT - 89T^{2} \)
97 \( 1 + 13.0iT - 97T^{2} \)
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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

−8.987872964583077640665090668771, −8.113920348105431063680857319242, −7.31280332165362520793642980416, −6.73567783533306374039538177332, −5.49417752491295549386840177820, −4.49570220829362880613138464039, −3.96044366146511209246563331698, −3.10894205498618576839974287452, −1.58031467269495285582942179155, −0.822653321893354989692276755470, 1.17674530576916794926980429009, 2.89501259007285069925821116668, 3.26425953136708848347629717289, 4.98251923665081197335647983128, 5.34035893658247277341595331547, 6.27679760885720840876861011397, 6.71693265527819204294962645441, 8.005551048956793530655034667427, 8.669795140289916146619880104285, 8.881523597815447015266046550546

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