Properties

Label 2-1400-7.4-c1-0-10
Degree $2$
Conductor $1400$
Sign $-0.266 - 0.963i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)3-s + (2 + 1.73i)7-s + (1 + 1.73i)9-s + (−1.5 + 2.59i)11-s + 6·13-s + (−2.5 + 4.33i)17-s + (−0.5 − 0.866i)19-s + (−2.5 + 0.866i)21-s + (−3.5 − 6.06i)23-s − 5·27-s + 2·29-s + (2.5 − 4.33i)31-s + (−1.5 − 2.59i)33-s + (1.5 + 2.59i)37-s + (−3 + 5.19i)39-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + (0.755 + 0.654i)7-s + (0.333 + 0.577i)9-s + (−0.452 + 0.783i)11-s + 1.66·13-s + (−0.606 + 1.05i)17-s + (−0.114 − 0.198i)19-s + (−0.545 + 0.188i)21-s + (−0.729 − 1.26i)23-s − 0.962·27-s + 0.371·29-s + (0.449 − 0.777i)31-s + (−0.261 − 0.452i)33-s + (0.246 + 0.427i)37-s + (−0.480 + 0.832i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $-0.266 - 0.963i$
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: \((2,\ 1400,\ (\ :1/2),\ -0.266 - 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.592580786\)
\(L(\frac12)\) \(\approx\) \(1.592580786\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2 - 1.73i)T \)
good3 \( 1 + (0.5 - 0.866i)T + (-1.5 - 2.59i)T^{2} \)
11 \( 1 + (1.5 - 2.59i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 6T + 13T^{2} \)
17 \( 1 + (2.5 - 4.33i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.5 + 6.06i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 + (-2.5 + 4.33i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.5 - 2.59i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + (-2.5 - 4.33i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (7.5 - 12.9i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.5 - 4.33i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.5 - 7.79i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (-3.5 + 6.06i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 12T + 83T^{2} \)
89 \( 1 + (3.5 + 6.06i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 2T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.987170380933182292777934332797, −8.814930063453752647490213279045, −8.343033125254692212301780658008, −7.52233515812876840574795672031, −6.27294652942890683990388644762, −5.71448049899111817187160039404, −4.51101996962661472736605195982, −4.21887654080069622407440767733, −2.56982473408329351555263787734, −1.58333758748498670802724269650, 0.71694095683985791390190510262, 1.70876384338215643278427297682, 3.31496916892761515790094972824, 4.10629585649288641949660291513, 5.24937498851628406251161061580, 6.11773983284387323927250460273, 6.84709508040139028640097800348, 7.72654194938336075444074332158, 8.404044813395472822934528611437, 9.248931491442342479376431271500

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