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Properties

Label 2-3344-76.75-c1-0-90
Degree $2$
Conductor $3344$
Sign $0.827 + 0.561i$
Analytic cond. $26.7019$
Root an. cond. $5.16739$
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  + 3.02·3-s + 2.92·5-s − 2.58i·7-s + 6.15·9-s − i·11-s − 1.48i·13-s + 8.84·15-s + 0.518·17-s + (−2.44 + 3.60i)19-s − 7.83i·21-s − 0.564i·23-s + 3.54·25-s + 9.56·27-s − 7.78i·29-s − 3.27·31-s + ⋯
L(s)  = 1  + 1.74·3-s + 1.30·5-s − 0.978i·7-s + 2.05·9-s − 0.301i·11-s − 0.413i·13-s + 2.28·15-s + 0.125·17-s + (−0.561 + 0.827i)19-s − 1.70i·21-s − 0.117i·23-s + 0.708·25-s + 1.84·27-s − 1.44i·29-s − 0.587·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.827 + 0.561i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.827 + 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3344\)    =    \(2^{4} \cdot 11 \cdot 19\)
Sign: $0.827 + 0.561i$
Analytic conductor: \(26.7019\)
Root analytic conductor: \(5.16739\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3344} (1519, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3344,\ (\ :1/2),\ 0.827 + 0.561i)\)

Particular Values

\(L(1)\) \(\approx\) \(4.655277975\)
\(L(\frac12)\) \(\approx\) \(4.655277975\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 + iT \)
19 \( 1 + (2.44 - 3.60i)T \)
good3 \( 1 - 3.02T + 3T^{2} \)
5 \( 1 - 2.92T + 5T^{2} \)
7 \( 1 + 2.58iT - 7T^{2} \)
13 \( 1 + 1.48iT - 13T^{2} \)
17 \( 1 - 0.518T + 17T^{2} \)
23 \( 1 + 0.564iT - 23T^{2} \)
29 \( 1 + 7.78iT - 29T^{2} \)
31 \( 1 + 3.27T + 31T^{2} \)
37 \( 1 + 5.18iT - 37T^{2} \)
41 \( 1 + 1.05iT - 41T^{2} \)
43 \( 1 - 4.86iT - 43T^{2} \)
47 \( 1 - 13.1iT - 47T^{2} \)
53 \( 1 - 6.61iT - 53T^{2} \)
59 \( 1 - 1.16T + 59T^{2} \)
61 \( 1 - 4.69T + 61T^{2} \)
67 \( 1 - 1.34T + 67T^{2} \)
71 \( 1 - 11.7T + 71T^{2} \)
73 \( 1 + 12.1T + 73T^{2} \)
79 \( 1 + 5.01T + 79T^{2} \)
83 \( 1 + 1.60iT - 83T^{2} \)
89 \( 1 - 12.5iT - 89T^{2} \)
97 \( 1 - 8.44iT - 97T^{2} \)
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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

−8.521711707460506035649706462082, −7.893183835321665013256261213525, −7.32946009908195220005505107928, −6.34554436377273857036708683199, −5.64681438654333887009645677537, −4.36980932154789328226100674704, −3.78070008631078821169699440399, −2.80622166178707643293489218241, −2.10911135025480372468871768210, −1.17942304547020325112484199113, 1.67185805720840534939514176271, 2.16808270627721716480399197750, 2.87641058299633035194706161908, 3.77148124964646565866622246528, 4.89610097412142100909902776878, 5.59423734697255437988108333827, 6.68407260815898654798732888983, 7.15796333026790983631520164698, 8.298236872667611154307282698368, 8.827221059641258134767853969547

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