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Properties

Label 2-3344-836.303-c0-0-0
Degree $2$
Conductor $3344$
Sign $-0.876 - 0.480i$
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  + (−0.564 + 1.73i)5-s + (0.169 − 0.122i)7-s + (−0.309 − 0.951i)9-s + (−0.104 + 0.994i)11-s + (−1.41 − 0.459i)17-s + (0.809 + 0.587i)19-s + 1.90i·23-s + (−1.89 − 1.37i)25-s + (0.118 + 0.363i)35-s − 1.95·43-s + 1.82·45-s + (0.244 − 0.336i)47-s + (−0.295 + 0.909i)49-s + (−1.66 − 0.743i)55-s + (0.395 + 0.128i)61-s + ⋯
L(s)  = 1  + (−0.564 + 1.73i)5-s + (0.169 − 0.122i)7-s + (−0.309 − 0.951i)9-s + (−0.104 + 0.994i)11-s + (−1.41 − 0.459i)17-s + (0.809 + 0.587i)19-s + 1.90i·23-s + (−1.89 − 1.37i)25-s + (0.118 + 0.363i)35-s − 1.95·43-s + 1.82·45-s + (0.244 − 0.336i)47-s + (−0.295 + 0.909i)49-s + (−1.66 − 0.743i)55-s + (0.395 + 0.128i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.876 - 0.480i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.876 - 0.480i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3344\)    =    \(2^{4} \cdot 11 \cdot 19\)
Sign: $-0.876 - 0.480i$
Analytic conductor: \(1.66887\)
Root analytic conductor: \(1.29184\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3344} (303, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3344,\ (\ :0),\ -0.876 - 0.480i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6767831652\)
\(L(\frac12)\) \(\approx\) \(0.6767831652\)
\(L(1)\) 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 \( 1 + (0.104 - 0.994i)T \)
19 \( 1 + (-0.809 - 0.587i)T \)
good3 \( 1 + (0.309 + 0.951i)T^{2} \)
5 \( 1 + (0.564 - 1.73i)T + (-0.809 - 0.587i)T^{2} \)
7 \( 1 + (-0.169 + 0.122i)T + (0.309 - 0.951i)T^{2} \)
13 \( 1 + (-0.809 + 0.587i)T^{2} \)
17 \( 1 + (1.41 + 0.459i)T + (0.809 + 0.587i)T^{2} \)
23 \( 1 - 1.90iT - T^{2} \)
29 \( 1 + (0.309 - 0.951i)T^{2} \)
31 \( 1 + (-0.809 + 0.587i)T^{2} \)
37 \( 1 + (-0.309 + 0.951i)T^{2} \)
41 \( 1 + (0.309 + 0.951i)T^{2} \)
43 \( 1 + 1.95T + T^{2} \)
47 \( 1 + (-0.244 + 0.336i)T + (-0.309 - 0.951i)T^{2} \)
53 \( 1 + (0.809 - 0.587i)T^{2} \)
59 \( 1 + (0.309 - 0.951i)T^{2} \)
61 \( 1 + (-0.395 - 0.128i)T + (0.809 + 0.587i)T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + (-0.809 - 0.587i)T^{2} \)
73 \( 1 + (1.01 + 1.40i)T + (-0.309 + 0.951i)T^{2} \)
79 \( 1 + (0.809 - 0.587i)T^{2} \)
83 \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.809 - 0.587i)T^{2} \)
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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

−9.316544906800215807356322511096, −8.186399603191011518847603370261, −7.41092161948813524368963609172, −6.97393984756347253400398489230, −6.36996866186200417115440031379, −5.41671108874240436769763898844, −4.29055352165265143412713395533, −3.51206492617318323793373370596, −2.89738243606557888317671927055, −1.79461359849255410039076567312, 0.38792409943835676314415812516, 1.70089646570925218635271323104, 2.80711404365138217518525547067, 4.03794874024182140718880009057, 4.76417709571320855459387892147, 5.19811492850546196788906682489, 6.10382676216844080341930199390, 7.09957496007053998745188380414, 8.125716550247771505765434437287, 8.587079894440288461408458172606

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