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Properties

Label 2-56e2-448.325-c0-0-0
Degree $2$
Conductor $3136$
Sign $0.682 - 0.731i$
Analytic cond. $1.56506$
Root an. cond. $1.25102$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.130 + 0.991i)2-s + (−0.965 + 0.258i)4-s + (−0.382 − 0.923i)8-s + (0.608 − 0.793i)9-s + (−0.389 + 0.0255i)11-s + (0.866 − 0.5i)16-s + (0.866 + 0.499i)18-s + (−0.0761 − 0.382i)22-s + (0.607 + 0.465i)23-s + (−0.793 + 0.608i)25-s + (1.08 − 1.63i)29-s + (0.608 + 0.793i)32-s + (−0.382 + 0.923i)36-s + (1.05 − 0.357i)37-s + (0.923 − 0.617i)43-s + (0.369 − 0.125i)44-s + ⋯
L(s)  = 1  + (0.130 + 0.991i)2-s + (−0.965 + 0.258i)4-s + (−0.382 − 0.923i)8-s + (0.608 − 0.793i)9-s + (−0.389 + 0.0255i)11-s + (0.866 − 0.5i)16-s + (0.866 + 0.499i)18-s + (−0.0761 − 0.382i)22-s + (0.607 + 0.465i)23-s + (−0.793 + 0.608i)25-s + (1.08 − 1.63i)29-s + (0.608 + 0.793i)32-s + (−0.382 + 0.923i)36-s + (1.05 − 0.357i)37-s + (0.923 − 0.617i)43-s + (0.369 − 0.125i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.682 - 0.731i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.682 - 0.731i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3136\)    =    \(2^{6} \cdot 7^{2}\)
Sign: $0.682 - 0.731i$
Analytic conductor: \(1.56506\)
Root analytic conductor: \(1.25102\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3136} (325, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3136,\ (\ :0),\ 0.682 - 0.731i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.217248767\)
\(L(\frac12)\) \(\approx\) \(1.217248767\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.130 - 0.991i)T \)
7 \( 1 \)
good3 \( 1 + (-0.608 + 0.793i)T^{2} \)
5 \( 1 + (0.793 - 0.608i)T^{2} \)
11 \( 1 + (0.389 - 0.0255i)T + (0.991 - 0.130i)T^{2} \)
13 \( 1 + (-0.923 + 0.382i)T^{2} \)
17 \( 1 + (0.866 - 0.5i)T^{2} \)
19 \( 1 + (-0.130 + 0.991i)T^{2} \)
23 \( 1 + (-0.607 - 0.465i)T + (0.258 + 0.965i)T^{2} \)
29 \( 1 + (-1.08 + 1.63i)T + (-0.382 - 0.923i)T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (-1.05 + 0.357i)T + (0.793 - 0.608i)T^{2} \)
41 \( 1 + (-0.707 + 0.707i)T^{2} \)
43 \( 1 + (-0.923 + 0.617i)T + (0.382 - 0.923i)T^{2} \)
47 \( 1 + (-0.866 - 0.5i)T^{2} \)
53 \( 1 + (-1.65 + 0.108i)T + (0.991 - 0.130i)T^{2} \)
59 \( 1 + (0.130 + 0.991i)T^{2} \)
61 \( 1 + (0.991 + 0.130i)T^{2} \)
67 \( 1 + (-1.49 + 0.735i)T + (0.608 - 0.793i)T^{2} \)
71 \( 1 + (0.541 - 1.30i)T + (-0.707 - 0.707i)T^{2} \)
73 \( 1 + (-0.965 - 0.258i)T^{2} \)
79 \( 1 + (0.478 - 1.78i)T + (-0.866 - 0.5i)T^{2} \)
83 \( 1 + (0.923 - 0.382i)T^{2} \)
89 \( 1 + (0.965 - 0.258i)T^{2} \)
97 \( 1 + T^{2} \)
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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

−8.845675241128042173005339121638, −8.082591357225575906325907096264, −7.39110127047762270709853302767, −6.77232755262498753119109785194, −5.96270792923324663433345699923, −5.33514520330423671436096203719, −4.28502965326860923427078197000, −3.78800026292294784140041152445, −2.58443403368221946378946083207, −0.915169952278502163906796899501, 1.12560839913382167008342505581, 2.25495432078859169523184930533, 2.98271519327317071594707331665, 4.10202230759479749519567682171, 4.74885662245761002995500806531, 5.43933575182331939552371498674, 6.43553868041069900341753879374, 7.44504648103400939828520078204, 8.172102781419189467611322548272, 8.862952506944209326459032486290

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