Properties

Label 2-3344-1.1-c1-0-1
Degree $2$
Conductor $3344$
Sign $1$
Analytic cond. $26.7019$
Root an. cond. $5.16739$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.129·3-s − 1.42·5-s − 5.10·7-s − 2.98·9-s − 11-s − 7.03·13-s + 0.183·15-s − 3.23·17-s + 19-s + 0.659·21-s + 6.34·23-s − 2.98·25-s + 0.774·27-s + 2.43·29-s − 8.51·31-s + 0.129·33-s + 7.24·35-s − 9.07·37-s + 0.909·39-s − 1.74·41-s − 2.02·43-s + 4.23·45-s + 5.67·47-s + 19.0·49-s + 0.418·51-s − 6.28·53-s + 1.42·55-s + ⋯
L(s)  = 1  − 0.0746·3-s − 0.635·5-s − 1.92·7-s − 0.994·9-s − 0.301·11-s − 1.95·13-s + 0.0474·15-s − 0.783·17-s + 0.229·19-s + 0.144·21-s + 1.32·23-s − 0.596·25-s + 0.148·27-s + 0.451·29-s − 1.52·31-s + 0.0225·33-s + 1.22·35-s − 1.49·37-s + 0.145·39-s − 0.272·41-s − 0.309·43-s + 0.631·45-s + 0.828·47-s + 2.71·49-s + 0.0585·51-s − 0.863·53-s + 0.191·55-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(0.1448556648\)
\(L(\frac12)\) \(\approx\) \(0.1448556648\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
19 \( 1 - T \)
good3 \( 1 + 0.129T + 3T^{2} \)
5 \( 1 + 1.42T + 5T^{2} \)
7 \( 1 + 5.10T + 7T^{2} \)
13 \( 1 + 7.03T + 13T^{2} \)
17 \( 1 + 3.23T + 17T^{2} \)
23 \( 1 - 6.34T + 23T^{2} \)
29 \( 1 - 2.43T + 29T^{2} \)
31 \( 1 + 8.51T + 31T^{2} \)
37 \( 1 + 9.07T + 37T^{2} \)
41 \( 1 + 1.74T + 41T^{2} \)
43 \( 1 + 2.02T + 43T^{2} \)
47 \( 1 - 5.67T + 47T^{2} \)
53 \( 1 + 6.28T + 53T^{2} \)
59 \( 1 - 1.86T + 59T^{2} \)
61 \( 1 - 4.98T + 61T^{2} \)
67 \( 1 + 4.01T + 67T^{2} \)
71 \( 1 + 4.62T + 71T^{2} \)
73 \( 1 + 13.9T + 73T^{2} \)
79 \( 1 - 7.19T + 79T^{2} \)
83 \( 1 - 9.72T + 83T^{2} \)
89 \( 1 - 15.1T + 89T^{2} \)
97 \( 1 + 6.23T + 97T^{2} \)
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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

−8.910252576805070467846234333862, −7.68778531729990357786879626236, −7.11124123964455268069640459807, −6.53114631762196918702573271813, −5.55333660615053258069944752695, −4.90444581742725674393215971926, −3.73342393259584133202822137080, −3.04378902255460495344846114525, −2.37210925642116337463351732076, −0.20601475513121320509638761630, 0.20601475513121320509638761630, 2.37210925642116337463351732076, 3.04378902255460495344846114525, 3.73342393259584133202822137080, 4.90444581742725674393215971926, 5.55333660615053258069944752695, 6.53114631762196918702573271813, 7.11124123964455268069640459807, 7.68778531729990357786879626236, 8.910252576805070467846234333862

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