Properties

Label 2-1400-7.4-c1-0-16
Degree $2$
Conductor $1400$
Sign $0.845 + 0.533i$
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  + (−1.64 + 2.84i)3-s + (−2.64 + 0.0641i)7-s + (−3.91 − 6.77i)9-s + (−2.91 + 5.04i)11-s − 2.75·13-s + (1 − 1.73i)17-s + (0.378 + 0.654i)19-s + (4.16 − 7.64i)21-s + (−0.266 − 0.462i)23-s + 15.8·27-s − 0.823·29-s + (1.28 − 2.23i)31-s + (−9.57 − 16.5i)33-s + (2.37 + 4.11i)37-s + (4.53 − 7.85i)39-s + ⋯
L(s)  = 1  + (−0.949 + 1.64i)3-s + (−0.999 + 0.0242i)7-s + (−1.30 − 2.25i)9-s + (−0.877 + 1.52i)11-s − 0.764·13-s + (0.242 − 0.420i)17-s + (0.0867 + 0.150i)19-s + (0.909 − 1.66i)21-s + (−0.0556 − 0.0963i)23-s + 3.05·27-s − 0.152·29-s + (0.231 − 0.401i)31-s + (−1.66 − 2.88i)33-s + (0.390 + 0.677i)37-s + (0.725 − 1.25i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.845 + 0.533i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.845 + 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $0.845 + 0.533i$
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: \((2,\ 1400,\ (\ :1/2),\ 0.845 + 0.533i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2247684787\)
\(L(\frac12)\) \(\approx\) \(0.2247684787\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.64 - 0.0641i)T \)
good3 \( 1 + (1.64 - 2.84i)T + (-1.5 - 2.59i)T^{2} \)
11 \( 1 + (2.91 - 5.04i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 2.75T + 13T^{2} \)
17 \( 1 + (-1 + 1.73i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.378 - 0.654i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.266 + 0.462i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 0.823T + 29T^{2} \)
31 \( 1 + (-1.28 + 2.23i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.37 - 4.11i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 6.06T + 41T^{2} \)
43 \( 1 + 0.710T + 43T^{2} \)
47 \( 1 + (6.44 + 11.1i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.20 - 7.27i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4 - 6.92i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.70 + 8.14i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.93 + 10.2i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (-1.75 + 3.04i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.75 - 8.23i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 6.71T + 83T^{2} \)
89 \( 1 + (0.878 + 1.52i)T + (-44.5 + 77.0i)T^{2} \)
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

−9.635337158298566857867368418216, −9.266966705371461394770983413282, −7.84562235975347527905367128661, −6.85350412606963416201693732872, −6.01320100145801899625368421896, −5.07648483758042023826063946338, −4.61592525990362564544253928966, −3.60105000481935211958106790521, −2.58967211230700265455543766319, −0.13286660692311196566955828435, 0.892076495874141906628092424764, 2.36956146489938941330991371180, 3.23344472917195347360603412962, 4.95473033171522449946738851238, 5.88790839308499388091146974366, 6.20036447095625564341582412658, 7.17561973300716944198653250778, 7.81046830523247981905453692415, 8.545394536290586648419249250957, 9.706043327531282922821372319459

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