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Properties

Label 2-1400-7.4-c1-0-34
Degree $2$
Conductor $1400$
Sign $-0.642 + 0.766i$
Analytic cond. $11.1790$
Root an. cond. $3.34350$
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  + (1.29 − 2.23i)3-s + (0.292 − 2.62i)7-s + (−1.83 − 3.18i)9-s + (−0.839 + 1.45i)11-s + 4.84·13-s + (1 − 1.73i)17-s + (−3.42 − 5.92i)19-s + (−5.50 − 4.05i)21-s + (−1.13 − 1.95i)23-s − 1.75·27-s + 3.32·29-s + (−4.58 + 7.94i)31-s + (2.16 + 3.75i)33-s + (−1.42 − 2.46i)37-s + (6.26 − 10.8i)39-s + ⋯
L(s)  = 1  + (0.746 − 1.29i)3-s + (0.110 − 0.993i)7-s + (−0.613 − 1.06i)9-s + (−0.253 + 0.438i)11-s + 1.34·13-s + (0.242 − 0.420i)17-s + (−0.785 − 1.36i)19-s + (−1.20 − 0.884i)21-s + (−0.235 − 0.408i)23-s − 0.337·27-s + 0.616·29-s + (−0.823 + 1.42i)31-s + (0.377 + 0.653i)33-s + (−0.233 − 0.405i)37-s + (1.00 − 1.73i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 + 0.766i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $-0.642 + 0.766i$
Analytic conductor: \(11.1790\)
Root analytic conductor: \(3.34350\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (1201, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1400,\ (\ :1/2),\ -0.642 + 0.766i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.098357223\)
\(L(\frac12)\) \(\approx\) \(2.098357223\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 + (-0.292 + 2.62i)T \)
good3 \( 1 + (-1.29 + 2.23i)T + (-1.5 - 2.59i)T^{2} \)
11 \( 1 + (0.839 - 1.45i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 4.84T + 13T^{2} \)
17 \( 1 + (-1 + 1.73i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.42 + 5.92i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.13 + 1.95i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 3.32T + 29T^{2} \)
31 \( 1 + (4.58 - 7.94i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.42 + 2.46i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 9.52T + 41T^{2} \)
43 \( 1 + 6.58T + 43T^{2} \)
47 \( 1 + (6.10 + 10.5i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.74 + 6.48i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4 - 6.92i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.24 - 5.62i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.87 - 4.98i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (5.84 - 10.1i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.84 + 4.93i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 12.5T + 83T^{2} \)
89 \( 1 + (-2.92 - 5.06i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 2T + 97T^{2} \)
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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

−8.846946959673896483337317665325, −8.511918622840745052506087829938, −7.53561445047202179869073489302, −6.96502504833044746150720774118, −6.39275949910265897150281935398, −5.04086334939984161663165727858, −3.95400089864543542881458976297, −2.93498547326164021996320962206, −1.84827669723851775842671044594, −0.794133202282664229731718095008, 1.82819906425089201051901301646, 3.08054158335646288331176212744, 3.75637787238653796427169407452, 4.61657161307370928158679932657, 5.78811320433008393150643069187, 6.18820326778911849300042075574, 8.019620566667289848147191687079, 8.252987552526407699304403131821, 9.145915266719472656419661595671, 9.657503006522604508110677148361

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