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Properties

Label 2-56e2-4.3-c0-0-6
Degree $2$
Conductor $3136$
Sign $i$
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  − i·3-s + 5-s − i·11-s − i·15-s − 17-s − i·19-s + i·23-s − i·27-s − i·31-s − 33-s + 37-s + i·47-s + i·51-s + 53-s − i·55-s + ⋯
L(s)  = 1  − i·3-s + 5-s − i·11-s − i·15-s − 17-s − i·19-s + i·23-s − i·27-s − i·31-s − 33-s + 37-s + i·47-s + i·51-s + 53-s − i·55-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3136\)    =    \(2^{6} \cdot 7^{2}\)
Sign: $i$
Analytic conductor: \(1.56506\)
Root analytic conductor: \(1.25102\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3136} (1471, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3136,\ (\ :0),\ i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.485652537\)
\(L(\frac12)\) \(\approx\) \(1.485652537\)
\(L(1)\) 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 \( 1 + iT - T^{2} \)
5 \( 1 - T + T^{2} \)
11 \( 1 + iT - T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + T + T^{2} \)
19 \( 1 + iT - T^{2} \)
23 \( 1 - iT - T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + iT - T^{2} \)
37 \( 1 - T + T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - iT - T^{2} \)
53 \( 1 - T + T^{2} \)
59 \( 1 + iT - T^{2} \)
61 \( 1 + T + T^{2} \)
67 \( 1 - iT - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T + T^{2} \)
79 \( 1 - iT - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - T + T^{2} \)
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

−8.672873226325139418633983288125, −7.83043602549637263353919826910, −7.16171680186417372324611151230, −6.30135256160612369572814490705, −5.96847750867158648624665783734, −4.98204821140025698768923604444, −3.94683850696249150388724611620, −2.70051482340089519087629051896, −2.02631818263215277663268418528, −0.940074708496143179824995466120, 1.65953217051652081927372094855, 2.48117009337847055691089189396, 3.70625648228680756649576216384, 4.49012310413206025673875988777, 5.05885138415428271470975104212, 5.99066298073151943228890797776, 6.69096785298671250857312614838, 7.52172645344500609831401430436, 8.562972428540158871598491415602, 9.241876990675967998884559216376

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