Properties

Label 2-1856-1.1-c3-0-80
Degree $2$
Conductor $1856$
Sign $1$
Analytic cond. $109.507$
Root an. cond. $10.4645$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.44·3-s + 19.6·5-s + 11.5·7-s − 24.8·9-s + 37.6·11-s + 44.5·13-s − 28.5·15-s − 61.1·17-s − 63.7·19-s − 16.8·21-s + 177.·23-s + 262.·25-s + 75.2·27-s − 29·29-s − 233.·31-s − 54.5·33-s + 228.·35-s − 10.2·37-s − 64.6·39-s + 347.·41-s + 194.·43-s − 490.·45-s − 14.5·47-s − 208.·49-s + 88.6·51-s + 606.·53-s + 741.·55-s + ⋯
L(s)  = 1  − 0.278·3-s + 1.76·5-s + 0.626·7-s − 0.922·9-s + 1.03·11-s + 0.951·13-s − 0.491·15-s − 0.873·17-s − 0.770·19-s − 0.174·21-s + 1.60·23-s + 2.10·25-s + 0.536·27-s − 0.185·29-s − 1.35·31-s − 0.287·33-s + 1.10·35-s − 0.0453·37-s − 0.265·39-s + 1.32·41-s + 0.690·43-s − 1.62·45-s − 0.0450·47-s − 0.607·49-s + 0.243·51-s + 1.57·53-s + 1.81·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1856\)    =    \(2^{6} \cdot 29\)
Sign: $1$
Analytic conductor: \(109.507\)
Root analytic conductor: \(10.4645\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1856} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1856,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(3.404917140\)
\(L(\frac12)\) \(\approx\) \(3.404917140\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
29 \( 1 + 29T \)
good3 \( 1 + 1.44T + 27T^{2} \)
5 \( 1 - 19.6T + 125T^{2} \)
7 \( 1 - 11.5T + 343T^{2} \)
11 \( 1 - 37.6T + 1.33e3T^{2} \)
13 \( 1 - 44.5T + 2.19e3T^{2} \)
17 \( 1 + 61.1T + 4.91e3T^{2} \)
19 \( 1 + 63.7T + 6.85e3T^{2} \)
23 \( 1 - 177.T + 1.21e4T^{2} \)
31 \( 1 + 233.T + 2.97e4T^{2} \)
37 \( 1 + 10.2T + 5.06e4T^{2} \)
41 \( 1 - 347.T + 6.89e4T^{2} \)
43 \( 1 - 194.T + 7.95e4T^{2} \)
47 \( 1 + 14.5T + 1.03e5T^{2} \)
53 \( 1 - 606.T + 1.48e5T^{2} \)
59 \( 1 - 702.T + 2.05e5T^{2} \)
61 \( 1 + 543.T + 2.26e5T^{2} \)
67 \( 1 - 407.T + 3.00e5T^{2} \)
71 \( 1 - 314.T + 3.57e5T^{2} \)
73 \( 1 + 859.T + 3.89e5T^{2} \)
79 \( 1 - 725.T + 4.93e5T^{2} \)
83 \( 1 - 820.T + 5.71e5T^{2} \)
89 \( 1 + 648.T + 7.04e5T^{2} \)
97 \( 1 + 60.9T + 9.12e5T^{2} \)
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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

−9.073610711614418324499108083097, −8.412495514299114916017531901731, −7.00556073560597870071801035131, −6.35809910755019332423044222800, −5.72742565749569061642830735776, −5.06715942564098621447472922715, −3.96169443977272417702898138007, −2.67342354536231267214817265604, −1.82960640149067584448321496804, −0.924704832233745873209505789863, 0.924704832233745873209505789863, 1.82960640149067584448321496804, 2.67342354536231267214817265604, 3.96169443977272417702898138007, 5.06715942564098621447472922715, 5.72742565749569061642830735776, 6.35809910755019332423044222800, 7.00556073560597870071801035131, 8.412495514299114916017531901731, 9.073610711614418324499108083097

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