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Properties

Label 2-1412-1412.1411-c0-0-6
Degree $2$
Conductor $1412$
Sign $1$
Analytic cond. $0.704679$
Root an. cond. $0.839452$
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  + 2-s + 1.41·3-s + 4-s + 1.41·6-s − 1.41·7-s + 8-s + 1.00·9-s + 1.41·12-s − 1.41·14-s + 16-s − 2·17-s + 1.00·18-s − 2.00·21-s + 1.41·24-s + 25-s − 1.41·28-s − 1.41·31-s + 32-s − 2·34-s + 1.00·36-s − 2.00·42-s + 1.41·48-s + 1.00·49-s + 50-s − 2.82·51-s − 1.41·56-s + 1.41·59-s + ⋯
L(s)  = 1  + 2-s + 1.41·3-s + 4-s + 1.41·6-s − 1.41·7-s + 8-s + 1.00·9-s + 1.41·12-s − 1.41·14-s + 16-s − 2·17-s + 1.00·18-s − 2.00·21-s + 1.41·24-s + 25-s − 1.41·28-s − 1.41·31-s + 32-s − 2·34-s + 1.00·36-s − 2.00·42-s + 1.41·48-s + 1.00·49-s + 50-s − 2.82·51-s − 1.41·56-s + 1.41·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1412 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1412 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1412\)    =    \(2^{2} \cdot 353\)
Sign: $1$
Analytic conductor: \(0.704679\)
Root analytic conductor: \(0.839452\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1412} (1411, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1412,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.658087394\)
\(L(\frac12)\) \(\approx\) \(2.658087394\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
353 \( 1 - T \)
good3 \( 1 - 1.41T + T^{2} \)
5 \( 1 - T^{2} \)
7 \( 1 + 1.41T + T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + 2T + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + 1.41T + T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 - 1.41T + T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 - 1.41T + T^{2} \)
71 \( 1 + 1.41T + T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - 1.41T + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + 2T + T^{2} \)
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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

−9.569568749220059068855373010091, −8.992777642670557168512939515592, −8.161235601154553648736112282172, −6.98730282094887284546760242655, −6.72708374155202454511857865653, −5.58090832885078982276496348297, −4.35337409978058536130211544522, −3.60940052259749619040295720701, −2.83702548198863313840827315371, −2.08606377780688723998127967378, 2.08606377780688723998127967378, 2.83702548198863313840827315371, 3.60940052259749619040295720701, 4.35337409978058536130211544522, 5.58090832885078982276496348297, 6.72708374155202454511857865653, 6.98730282094887284546760242655, 8.161235601154553648736112282172, 8.992777642670557168512939515592, 9.569568749220059068855373010091

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