Properties

Label 2-56e2-112.61-c0-0-0
Degree $2$
Conductor $3136$
Sign $-0.897 + 0.440i$
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 + (−1.36 − 0.366i)11-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 + (−0.366 + 1.36i)67-s − 2i·71-s + (0.499 + 0.866i)81-s + (0.999 + i)99-s + (−0.366 − 1.36i)107-s + (0.366 − 1.36i)109-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)9-s + (−1.36 − 0.366i)11-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 + (−0.366 + 1.36i)67-s − 2i·71-s + (0.499 + 0.866i)81-s + (0.999 + i)99-s + (−0.366 − 1.36i)107-s + (0.366 − 1.36i)109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.897 + 0.440i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.897 + 0.440i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3136\)    =    \(2^{6} \cdot 7^{2}\)
Sign: $-0.897 + 0.440i$
Analytic conductor: \(1.56506\)
Root analytic conductor: \(1.25102\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3136} (1489, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3136,\ (\ :0),\ -0.897 + 0.440i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3199560453\)
\(L(\frac12)\) \(\approx\) \(0.3199560453\)
\(L(1)\) 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 \( 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.866 + 0.5i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
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 \( 1 + (0.366 - 1.36i)T + (-0.866 - 0.5i)T^{2} \)
71 \( 1 + 2iT - T^{2} \)
73 \( 1 + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T^{2} \)
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

−8.427996418459638467057432088843, −7.900512930318205910375536823986, −7.16738523041969537860649366702, −6.02657825756519963457876702069, −5.67309401242918886976098771410, −4.77054521313406216804833145599, −3.66167939576671648536203342340, −2.93313081480110696234522764150, −1.94836587209858076261051358068, −0.17520951856914355520551977233, 1.82753241776595860201490764989, 2.71081471819729334289083109492, 3.55142938201465429828074858978, 4.79171495365004877020756195307, 5.29139649609443262546872212072, 6.06959998955496526407695458299, 7.03404360216599192855688667931, 7.83994779585751711375043450142, 8.293296188471047849945881907972, 9.114815122480079394975096333348

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