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Properties

Label 2-56e2-112.51-c0-0-0
Degree $2$
Conductor $3136$
Sign $0.254 - 0.966i$
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  + (−0.866 − 0.5i)9-s + (−0.366 + 1.36i)11-s + (−1 + 1.73i)23-s + (0.866 − 0.5i)25-s + (1 + i)29-s + (0.366 + 1.36i)37-s + (1 + i)43-s + (−1.36 − 0.366i)53-s + (1.36 + 0.366i)67-s + (−1.73 − i)79-s + (0.499 + 0.866i)81-s + (1 − 0.999i)99-s + (1.36 − 0.366i)107-s + (−0.366 + 1.36i)109-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)9-s + (−0.366 + 1.36i)11-s + (−1 + 1.73i)23-s + (0.866 − 0.5i)25-s + (1 + i)29-s + (0.366 + 1.36i)37-s + (1 + i)43-s + (−1.36 − 0.366i)53-s + (1.36 + 0.366i)67-s + (−1.73 − i)79-s + (0.499 + 0.866i)81-s + (1 − 0.999i)99-s + (1.36 − 0.366i)107-s + (−0.366 + 1.36i)109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.254 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.254 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3136\)    =    \(2^{6} \cdot 7^{2}\)
Sign: $0.254 - 0.966i$
Analytic conductor: \(1.56506\)
Root analytic conductor: \(1.25102\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3136} (1647, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3136,\ (\ :0),\ 0.254 - 0.966i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9555322539\)
\(L(\frac12)\) \(\approx\) \(0.9555322539\)
\(L(1)\) 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 \( 1 + (0.866 + 0.5i)T^{2} \)
5 \( 1 + (-0.866 + 0.5i)T^{2} \)
11 \( 1 + (0.366 - 1.36i)T + (-0.866 - 0.5i)T^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.866 + 0.5i)T^{2} \)
23 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (-1 - i)T + iT^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + (-1 - i)T + iT^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2} \)
59 \( 1 + (-0.866 - 0.5i)T^{2} \)
61 \( 1 + (0.866 - 0.5i)T^{2} \)
67 \( 1 + (-1.36 - 0.366i)T + (0.866 + 0.5i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (1.73 + i)T + (0.5 + 0.866i)T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
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

−9.055454888740109864935558763768, −8.233182975542733848226096295840, −7.58870429447240524747418824385, −6.74819956544759876393350815165, −6.05699670365511323374400575283, −5.13926342527553665391787995648, −4.47559466961476429248012606820, −3.38348978933619090248109067271, −2.59112895467443584356043755052, −1.41646290123275815725696172388, 0.59244167575062432436838804239, 2.31355278674019316467908035807, 2.93958174251588638395214066521, 3.99802879471482290362780376973, 4.91773627386179798773190807724, 5.81496394181867140380969245686, 6.22461487432728380234421541028, 7.28442729504375339708428731681, 8.314372507895499622887003895409, 8.402021085623225893175264708347

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