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Properties

Label 2-1400-35.13-c1-0-11
Degree $2$
Conductor $1400$
Sign $0.431 - 0.902i$
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  + (−0.0703 + 0.0703i)3-s + (0.562 − 2.58i)7-s + 2.99i·9-s + 0.777·11-s + (−3.93 + 3.93i)13-s + (0.982 + 0.982i)17-s + 1.14·19-s + (0.142 + 0.221i)21-s + (1.46 + 1.46i)23-s + (−0.421 − 0.421i)27-s + 4.69i·29-s + 6.45i·31-s + (−0.0546 + 0.0546i)33-s + (1.30 − 1.30i)37-s − 0.553i·39-s + ⋯
L(s)  = 1  + (−0.0405 + 0.0405i)3-s + (0.212 − 0.977i)7-s + 0.996i·9-s + 0.234·11-s + (−1.09 + 1.09i)13-s + (0.238 + 0.238i)17-s + 0.263·19-s + (0.0310 + 0.0482i)21-s + (0.304 + 0.304i)23-s + (−0.0810 − 0.0810i)27-s + 0.870i·29-s + 1.15i·31-s + (−0.00951 + 0.00951i)33-s + (0.214 − 0.214i)37-s − 0.0886i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.431 - 0.902i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.431 - 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $0.431 - 0.902i$
Analytic conductor: \(11.1790\)
Root analytic conductor: \(3.34350\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (993, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1400,\ (\ :1/2),\ 0.431 - 0.902i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.437671413\)
\(L(\frac12)\) \(\approx\) \(1.437671413\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.562 + 2.58i)T \)
good3 \( 1 + (0.0703 - 0.0703i)T - 3iT^{2} \)
11 \( 1 - 0.777T + 11T^{2} \)
13 \( 1 + (3.93 - 3.93i)T - 13iT^{2} \)
17 \( 1 + (-0.982 - 0.982i)T + 17iT^{2} \)
19 \( 1 - 1.14T + 19T^{2} \)
23 \( 1 + (-1.46 - 1.46i)T + 23iT^{2} \)
29 \( 1 - 4.69iT - 29T^{2} \)
31 \( 1 - 6.45iT - 31T^{2} \)
37 \( 1 + (-1.30 + 1.30i)T - 37iT^{2} \)
41 \( 1 + 9.81iT - 41T^{2} \)
43 \( 1 + (-7.13 - 7.13i)T + 43iT^{2} \)
47 \( 1 + (-7.34 - 7.34i)T + 47iT^{2} \)
53 \( 1 + (-2.08 - 2.08i)T + 53iT^{2} \)
59 \( 1 + 8.29T + 59T^{2} \)
61 \( 1 - 5.88iT - 61T^{2} \)
67 \( 1 + (-6.30 + 6.30i)T - 67iT^{2} \)
71 \( 1 - 12.3T + 71T^{2} \)
73 \( 1 + (-7.23 + 7.23i)T - 73iT^{2} \)
79 \( 1 - 2.83iT - 79T^{2} \)
83 \( 1 + (10.4 - 10.4i)T - 83iT^{2} \)
89 \( 1 + 3.41T + 89T^{2} \)
97 \( 1 + (-8.50 - 8.50i)T + 97iT^{2} \)
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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

−9.665274407500989472434138740633, −9.031178795583720435468660499318, −7.88763566162685557607751558300, −7.33827673006812556752057571923, −6.64877213962514989527746819691, −5.35459729009407390624755796347, −4.66047090919843146985386490878, −3.82229802692041385203912526396, −2.49886637012997781609619319405, −1.33282012519571227197449963548, 0.62816243819361895433466850009, 2.30610303347765458208087668563, 3.15007855760757494151390474492, 4.34100589512947223007893804750, 5.40220845457887361202417858020, 5.98029745300737103503890919931, 6.99173784431379110582880937113, 7.85971689227671600047107629518, 8.624560433269808060949739429715, 9.571609654194785503726209856455

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