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Properties

Label 2-2106-13.12-c1-0-15
Degree $2$
Conductor $2106$
Sign $0.542 - 0.839i$
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 + 2.77i·5-s + 0.876i·7-s + i·8-s + 2.77·10-s − 2.22i·11-s + (3.02 + 1.95i)13-s + 0.876·14-s + 16-s + 3.80·17-s + 2.22i·19-s − 2.77i·20-s − 2.22·22-s + 0.519·23-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s + 1.24i·5-s + 0.331i·7-s + 0.353i·8-s + 0.878·10-s − 0.671i·11-s + (0.839 + 0.542i)13-s + 0.234·14-s + 0.250·16-s + 0.923·17-s + 0.510i·19-s − 0.621i·20-s − 0.474·22-s + 0.108·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2106 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.542 - 0.839i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2106 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.542 - 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2106\)    =    \(2 \cdot 3^{4} \cdot 13\)
Sign: $0.542 - 0.839i$
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: \((2,\ 2106,\ (\ :1/2),\ 0.542 - 0.839i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.483942367\)
\(L(\frac12)\) \(\approx\) \(1.483942367\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.02 - 1.95i)T \)
good5 \( 1 - 2.77iT - 5T^{2} \)
7 \( 1 - 0.876iT - 7T^{2} \)
11 \( 1 + 2.22iT - 11T^{2} \)
17 \( 1 - 3.80T + 17T^{2} \)
19 \( 1 - 2.22iT - 19T^{2} \)
23 \( 1 - 0.519T + 23T^{2} \)
29 \( 1 + 7.62T + 29T^{2} \)
31 \( 1 + 5.74iT - 31T^{2} \)
37 \( 1 - 11.3iT - 37T^{2} \)
41 \( 1 - 4.06iT - 41T^{2} \)
43 \( 1 + 5.63T + 43T^{2} \)
47 \( 1 - 1.06iT - 47T^{2} \)
53 \( 1 + 7.29T + 53T^{2} \)
59 \( 1 + 4.06iT - 59T^{2} \)
61 \( 1 - 7.89T + 61T^{2} \)
67 \( 1 - 6.87iT - 67T^{2} \)
71 \( 1 - 15.9iT - 71T^{2} \)
73 \( 1 - 5.24iT - 73T^{2} \)
79 \( 1 - 14.1T + 79T^{2} \)
83 \( 1 + 0.740iT - 83T^{2} \)
89 \( 1 + 9.89iT - 89T^{2} \)
97 \( 1 - 15.7iT - 97T^{2} \)
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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

−9.443731582954631856182165218060, −8.452419426081855352471188976974, −7.83780149269759760547508241258, −6.77854739677429305089023688964, −6.08079854860931886173835834903, −5.29783973908884187652771194533, −3.94374990154309578200256120353, −3.35519152694320375887498321763, −2.51381975290325287023175468866, −1.33231402080072245719313731567, 0.57454855781378304451277422801, 1.70679614174622875374804064353, 3.42123893188990892949279818063, 4.23138585005864836368462534021, 5.19263809418830435819309856661, 5.58386191685076431824584602123, 6.69876620984418544291464142058, 7.53371992452847502847162697357, 8.117721986347556144450049459779, 9.011602086521763764244778214467

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