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Properties

Label 2-3344-836.791-c0-0-0
Degree $2$
Conductor $3344$
Sign $-0.0977 + 0.995i$
Analytic cond. $1.66887$
Root an. cond. $1.29184$
Motivic weight $0$
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)5-s − 7-s + (0.5 − 0.866i)9-s + 11-s + (0.5 + 0.866i)19-s + (0.5 + 0.866i)35-s − 1.73i·37-s + (−1 − 1.73i)43-s − 0.999·45-s + (−1.5 − 0.866i)53-s + (−0.5 − 0.866i)55-s + (−0.5 + 0.866i)63-s − 77-s + (1.5 − 0.866i)79-s + (−0.499 − 0.866i)81-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)5-s − 7-s + (0.5 − 0.866i)9-s + 11-s + (0.5 + 0.866i)19-s + (0.5 + 0.866i)35-s − 1.73i·37-s + (−1 − 1.73i)43-s − 0.999·45-s + (−1.5 − 0.866i)53-s + (−0.5 − 0.866i)55-s + (−0.5 + 0.866i)63-s − 77-s + (1.5 − 0.866i)79-s + (−0.499 − 0.866i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0977 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0977 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3344\)    =    \(2^{4} \cdot 11 \cdot 19\)
Sign: $-0.0977 + 0.995i$
Analytic conductor: \(1.66887\)
Root analytic conductor: \(1.29184\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3344} (2463, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3344,\ (\ :0),\ -0.0977 + 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9824886054\)
\(L(\frac12)\) \(\approx\) \(0.9824886054\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 - T \)
19 \( 1 + (-0.5 - 0.866i)T \)
good3 \( 1 + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
7 \( 1 + T + T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + 1.73iT - T^{2} \)
41 \( 1 + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.5 + 0.866i)T^{2} \)
73 \( 1 + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
83 \( 1 + T + T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
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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

−8.781709953879737035262689963189, −7.893658345588534422232579627636, −7.02186097365728615672213171817, −6.43345799203336557288264208870, −5.66319321749649679007861006794, −4.63596351113743426226949989137, −3.76412889198418440304219515925, −3.40762661930304385902763075025, −1.79545191185697801519104198143, −0.62839135175571514933065219040, 1.44126639225351194546092073727, 2.82994591263405118069470157563, 3.32729503557027322658926649766, 4.33183717411393938849532959941, 5.09324899292741106140592203599, 6.40826105172078709610623400627, 6.63350743497419124202106247715, 7.45191202393599930272302479161, 8.106058885971312147726734417527, 9.131009534053220939561174464906

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