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Properties

Label 2-56e2-8.5-c1-0-28
Degree $2$
Conductor $3136$
Sign $-0.707 - 0.707i$
Analytic cond. $25.0410$
Root an. cond. $5.00410$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2i·3-s − 9-s + 6i·11-s + 6·17-s + 2i·19-s + 5·25-s + 4i·27-s − 12·33-s + 6·41-s − 10i·43-s + 12i·51-s − 4·57-s − 6i·59-s + 14i·67-s − 2·73-s + ⋯
L(s)  = 1  + 1.15i·3-s − 0.333·9-s + 1.80i·11-s + 1.45·17-s + 0.458i·19-s + 25-s + 0.769i·27-s − 2.08·33-s + 0.937·41-s − 1.52i·43-s + 1.68i·51-s − 0.529·57-s − 0.781i·59-s + 1.71i·67-s − 0.234·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3136\)    =    \(2^{6} \cdot 7^{2}\)
Sign: $-0.707 - 0.707i$
Analytic conductor: \(25.0410\)
Root analytic conductor: \(5.00410\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3136} (1569, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3136,\ (\ :1/2),\ -0.707 - 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.904129867\)
\(L(\frac12)\) \(\approx\) \(1.904129867\)
\(L(\frac{3}{2})\) 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 \( 1 - 2iT - 3T^{2} \)
5 \( 1 - 5T^{2} \)
11 \( 1 - 6iT - 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 - 6T + 17T^{2} \)
19 \( 1 - 2iT - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 + 10iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 6iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 14iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 2T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 18iT - 83T^{2} \)
89 \( 1 + 18T + 89T^{2} \)
97 \( 1 + 10T + 97T^{2} \)
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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

−9.201300573582474612375567317995, −8.301216713866259865374783361216, −7.39638960107540757648478094970, −6.88931944157211254236916594557, −5.63857653717178333625585159770, −5.05486790880459989757002431341, −4.28173620210218781996009261464, −3.65704973096398465636691511906, −2.57541598618848088917946280006, −1.37057478058171421613210447925, 0.66636582296721380681982437753, 1.38475969050063167832128768248, 2.75599499978040269243649258458, 3.34624192029341606392474430272, 4.57089174409361028415303902823, 5.67578685225208399490778188244, 6.11992261371291265588750545832, 6.94679667565184657506261782776, 7.70500045855075367769395874182, 8.232360236909202413306296243476

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