Properties

Label 2-1400-5.4-c1-0-10
Degree $2$
Conductor $1400$
Sign $-0.447 - 0.894i$
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  + 2.37i·3-s + i·7-s − 2.62·9-s + 6.37·11-s + 4.37i·13-s + 0.372i·17-s + 4.74·19-s − 2.37·21-s − 4.74i·23-s + 0.883i·27-s + 4.37·29-s − 8·31-s + 15.1i·33-s + 2i·37-s − 10.3·39-s + ⋯
L(s)  = 1  + 1.36i·3-s + 0.377i·7-s − 0.875·9-s + 1.92·11-s + 1.21i·13-s + 0.0902i·17-s + 1.08·19-s − 0.517·21-s − 0.989i·23-s + 0.169i·27-s + 0.811·29-s − 1.43·31-s + 2.63i·33-s + 0.328i·37-s − 1.66·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $-0.447 - 0.894i$
Analytic conductor: \(11.1790\)
Root analytic conductor: \(3.34350\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1400,\ (\ :1/2),\ -0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.874969935\)
\(L(\frac12)\) \(\approx\) \(1.874969935\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
good3 \( 1 - 2.37iT - 3T^{2} \)
11 \( 1 - 6.37T + 11T^{2} \)
13 \( 1 - 4.37iT - 13T^{2} \)
17 \( 1 - 0.372iT - 17T^{2} \)
19 \( 1 - 4.74T + 19T^{2} \)
23 \( 1 + 4.74iT - 23T^{2} \)
29 \( 1 - 4.37T + 29T^{2} \)
31 \( 1 + 8T + 31T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 - 6.74T + 41T^{2} \)
43 \( 1 + 8.74iT - 43T^{2} \)
47 \( 1 - 7.11iT - 47T^{2} \)
53 \( 1 - 10.7iT - 53T^{2} \)
59 \( 1 + 8T + 59T^{2} \)
61 \( 1 + 2.74T + 61T^{2} \)
67 \( 1 - 4iT - 67T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 + 6iT - 73T^{2} \)
79 \( 1 + 15.1T + 79T^{2} \)
83 \( 1 + 9.48iT - 83T^{2} \)
89 \( 1 + 14.7T + 89T^{2} \)
97 \( 1 - 9.86iT - 97T^{2} \)
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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

−9.537271975139311937829241812424, −9.226197626728592459924967430238, −8.681249870404914166408211031481, −7.29189226638239601667964196034, −6.47065943547911520903316357830, −5.59314833508952772011714162803, −4.44244714552740709788916182633, −4.09104890284744200222368086986, −3.04084918377768325476006952003, −1.51379402837320561826136960514, 0.878826684662716333999914239061, 1.63614467732179163470468212709, 3.07609754494843079586119052947, 3.98835724350761741555531399266, 5.36300106248411565105561122905, 6.19361746890550661782612452121, 6.98272888563043599793340005416, 7.52459587519560767090339135456, 8.294451067605778344873146695957, 9.285636565483264265757542536053

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