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Properties

Label 2-56e2-28.27-c1-0-64
Degree $2$
Conductor $3136$
Sign $0.755 + 0.654i$
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  + 2.64·3-s + 1.73i·5-s + 4.00·9-s − 4.58i·11-s − 3.46i·13-s + 4.58i·15-s − 5.19i·17-s + 2.64·19-s − 4.58i·23-s + 2.00·25-s + 2.64·27-s + 2.64·31-s − 12.1i·33-s − 7·37-s − 9.16i·39-s + ⋯
L(s)  = 1  + 1.52·3-s + 0.774i·5-s + 1.33·9-s − 1.38i·11-s − 0.960i·13-s + 1.18i·15-s − 1.26i·17-s + 0.606·19-s − 0.955i·23-s + 0.400·25-s + 0.509·27-s + 0.475·31-s − 2.11i·33-s − 1.15·37-s − 1.46i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 + 0.654i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.755 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3136\)    =    \(2^{6} \cdot 7^{2}\)
Sign: $0.755 + 0.654i$
Analytic conductor: \(25.0410\)
Root analytic conductor: \(5.00410\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3136} (3135, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3136,\ (\ :1/2),\ 0.755 + 0.654i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.209523946\)
\(L(\frac12)\) \(\approx\) \(3.209523946\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.64T + 3T^{2} \)
5 \( 1 - 1.73iT - 5T^{2} \)
11 \( 1 + 4.58iT - 11T^{2} \)
13 \( 1 + 3.46iT - 13T^{2} \)
17 \( 1 + 5.19iT - 17T^{2} \)
19 \( 1 - 2.64T + 19T^{2} \)
23 \( 1 + 4.58iT - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 2.64T + 31T^{2} \)
37 \( 1 + 7T + 37T^{2} \)
41 \( 1 + 3.46iT - 41T^{2} \)
43 \( 1 - 9.16iT - 43T^{2} \)
47 \( 1 - 7.93T + 47T^{2} \)
53 \( 1 + 3T + 53T^{2} \)
59 \( 1 + 7.93T + 59T^{2} \)
61 \( 1 + 1.73iT - 61T^{2} \)
67 \( 1 - 4.58iT - 67T^{2} \)
71 \( 1 + 9.16iT - 71T^{2} \)
73 \( 1 - 5.19iT - 73T^{2} \)
79 \( 1 + 4.58iT - 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 - 1.73iT - 89T^{2} \)
97 \( 1 + 3.46iT - 97T^{2} \)
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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

−8.629780564372490924653068491675, −7.900140346620391965295530184563, −7.34645336525681395231575104436, −6.50411872635940670815663184263, −5.59365762073194727252809089774, −4.58176904185654472277630225274, −3.24336696829549375517716097413, −3.18777273747115467427447212136, −2.36061530168356525117958321278, −0.818352477020158837121686251864, 1.52188120599330435962466241390, 2.02724369103415553441576227845, 3.19819162945482205741633412065, 4.05191630937557167829505329873, 4.63339741888156785238105259158, 5.62929254218544884908710262072, 6.85250206414875499767101432718, 7.39837272794117354484449337629, 8.178063747961099898750728907194, 8.796017740608884446579843229976

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