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Properties

Label 2-56e2-4.3-c0-0-2
Degree $2$
Conductor $3136$
Sign $1$
Analytic cond. $1.56506$
Root an. cond. $1.25102$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.41·5-s + 9-s + 1.41·13-s − 1.41·17-s + 1.00·25-s + 1.41·41-s − 1.41·45-s + 2·53-s + 1.41·61-s − 2.00·65-s + 1.41·73-s + 81-s + 2.00·85-s − 1.41·89-s + 1.41·97-s + 1.41·101-s − 2·113-s + 1.41·117-s + ⋯
L(s)  = 1  − 1.41·5-s + 9-s + 1.41·13-s − 1.41·17-s + 1.00·25-s + 1.41·41-s − 1.41·45-s + 2·53-s + 1.41·61-s − 2.00·65-s + 1.41·73-s + 81-s + 2.00·85-s − 1.41·89-s + 1.41·97-s + 1.41·101-s − 2·113-s + 1.41·117-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3136\)    =    \(2^{6} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(1.56506\)
Root analytic conductor: \(1.25102\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3136} (1471, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3136,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.050593243\)
\(L(\frac12)\) \(\approx\) \(1.050593243\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 - T^{2} \)
5 \( 1 + 1.41T + T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 - 1.41T + T^{2} \)
17 \( 1 + 1.41T + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 - 1.41T + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - 2T + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - 1.41T + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - 1.41T + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + 1.41T + T^{2} \)
97 \( 1 - 1.41T + T^{2} \)
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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

−8.682625522518993230230418290539, −8.191283829371242140222781063716, −7.32510254339172346950589462928, −6.81445519294752452064862739640, −5.94535708777999375356248951104, −4.76358792281396127746686229471, −4.03537318033942660351037788810, −3.65549184574613610309833093417, −2.29996411623702513081366619020, −0.946465387379902088889797173362, 0.946465387379902088889797173362, 2.29996411623702513081366619020, 3.65549184574613610309833093417, 4.03537318033942660351037788810, 4.76358792281396127746686229471, 5.94535708777999375356248951104, 6.81445519294752452064862739640, 7.32510254339172346950589462928, 8.191283829371242140222781063716, 8.682625522518993230230418290539

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