Properties

Label 2-56e2-1.1-c1-0-57
Degree $2$
Conductor $3136$
Sign $-1$
Analytic cond. $25.0410$
Root an. cond. $5.00410$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·9-s + 4·11-s − 8·23-s − 5·25-s − 2·29-s + 6·37-s − 12·43-s + 10·53-s + 4·67-s − 16·71-s − 8·79-s + 9·81-s − 12·99-s − 20·107-s − 18·109-s + 2·113-s + ⋯
L(s)  = 1  − 9-s + 1.20·11-s − 1.66·23-s − 25-s − 0.371·29-s + 0.986·37-s − 1.82·43-s + 1.37·53-s + 0.488·67-s − 1.89·71-s − 0.900·79-s + 81-s − 1.20·99-s − 1.93·107-s − 1.72·109-s + 0.188·113-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3136\)    =    \(2^{6} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(25.0410\)
Root analytic conductor: \(5.00410\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{3136} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 3136,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + p T^{2} \)
5 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + p T^{2} \)
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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

−8.330759812635606547523055682566, −7.68998546052449448134866019451, −6.67625625855292974207346060772, −6.04800902507783081318745570011, −5.40928035513183831948820848756, −4.22989383620026669877623350668, −3.65330783308024527411814200780, −2.54910322548506501414462221460, −1.53416694723259152452558705643, 0, 1.53416694723259152452558705643, 2.54910322548506501414462221460, 3.65330783308024527411814200780, 4.22989383620026669877623350668, 5.40928035513183831948820848756, 6.04800902507783081318745570011, 6.67625625855292974207346060772, 7.68998546052449448134866019451, 8.330759812635606547523055682566

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