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Properties

Label 2-1400-7.2-c1-0-34
Degree $2$
Conductor $1400$
Sign $-0.701 - 0.712i$
Analytic cond. $11.1790$
Root an. cond. $3.34350$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 − 1.73i)3-s + (0.5 + 2.59i)7-s + (−0.499 + 0.866i)9-s + (−2 − 3.46i)11-s − 2·13-s + (1.5 + 2.59i)17-s + (4 − 3.46i)21-s + (−1.5 + 2.59i)23-s − 4.00·27-s − 6·29-s + (−4.5 − 7.79i)31-s + (−3.99 + 6.92i)33-s + (2 + 3.46i)39-s + 5·41-s − 6·43-s + ⋯
L(s)  = 1  + (−0.577 − 0.999i)3-s + (0.188 + 0.981i)7-s + (−0.166 + 0.288i)9-s + (−0.603 − 1.04i)11-s − 0.554·13-s + (0.363 + 0.630i)17-s + (0.872 − 0.755i)21-s + (−0.312 + 0.541i)23-s − 0.769·27-s − 1.11·29-s + (−0.808 − 1.39i)31-s + (−0.696 + 1.20i)33-s + (0.320 + 0.554i)39-s + 0.780·41-s − 0.914·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.701 - 0.712i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.701 - 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $-0.701 - 0.712i$
Analytic conductor: \(11.1790\)
Root analytic conductor: \(3.34350\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (401, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 1400,\ (\ :1/2),\ -0.701 - 0.712i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 2.59i)T \)
good3 \( 1 + (1 + 1.73i)T + (-1.5 + 2.59i)T^{2} \)
11 \( 1 + (2 + 3.46i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 2T + 13T^{2} \)
17 \( 1 + (-1.5 - 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.5 - 2.59i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + (4.5 + 7.79i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 5T + 41T^{2} \)
43 \( 1 + 6T + 43T^{2} \)
47 \( 1 + (4.5 - 7.79i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3 - 5.19i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (4 + 6.92i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4 - 6.92i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (7 + 12.1i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 11T + 71T^{2} \)
73 \( 1 + (-1 - 1.73i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.5 - 7.79i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 6T + 83T^{2} \)
89 \( 1 + (5.5 - 9.52i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 11T + 97T^{2} \)
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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

−9.057584356205617008849450695847, −7.930979535939293679161368586035, −7.64793708310325999463619622478, −6.39602394566470152406455152396, −5.82399844926110505481187087409, −5.23782508294502488691160452622, −3.77722340681082264720667633945, −2.57651444575927151677618964359, −1.53773469734229916152835142090, 0, 1.87316942471451624229064337781, 3.33682854025418163583042462484, 4.37244294878825834841474133860, 4.89823152790279610510103485348, 5.64893301469226023347863315244, 7.07545341765656094034780729919, 7.39391988336144197096625548699, 8.506399334216280095394423175577, 9.755723062234283288894935282347

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