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Properties

Label 2-1400-35.27-c1-0-30
Degree $2$
Conductor $1400$
Sign $0.492 + 0.870i$
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  + (1.26 + 1.26i)3-s + (−1.98 − 1.75i)7-s + 0.201i·9-s + 4.28·11-s + (−4.40 − 4.40i)13-s + (3.77 − 3.77i)17-s − 8.01·19-s + (−0.292 − 4.72i)21-s + (2.07 − 2.07i)23-s + (3.54 − 3.54i)27-s + 0.383i·29-s − 1.01i·31-s + (5.41 + 5.41i)33-s + (−5.30 − 5.30i)37-s − 11.1i·39-s + ⋯
L(s)  = 1  + (0.730 + 0.730i)3-s + (−0.749 − 0.662i)7-s + 0.0670i·9-s + 1.29·11-s + (−1.22 − 1.22i)13-s + (0.914 − 0.914i)17-s − 1.83·19-s + (−0.0638 − 1.03i)21-s + (0.432 − 0.432i)23-s + (0.681 − 0.681i)27-s + 0.0712i·29-s − 0.182i·31-s + (0.942 + 0.942i)33-s + (−0.871 − 0.871i)37-s − 1.78i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.492 + 0.870i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.492 + 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $0.492 + 0.870i$
Analytic conductor: \(11.1790\)
Root analytic conductor: \(3.34350\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (657, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1400,\ (\ :1/2),\ 0.492 + 0.870i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.665075827\)
\(L(\frac12)\) \(\approx\) \(1.665075827\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.98 + 1.75i)T \)
good3 \( 1 + (-1.26 - 1.26i)T + 3iT^{2} \)
11 \( 1 - 4.28T + 11T^{2} \)
13 \( 1 + (4.40 + 4.40i)T + 13iT^{2} \)
17 \( 1 + (-3.77 + 3.77i)T - 17iT^{2} \)
19 \( 1 + 8.01T + 19T^{2} \)
23 \( 1 + (-2.07 + 2.07i)T - 23iT^{2} \)
29 \( 1 - 0.383iT - 29T^{2} \)
31 \( 1 + 1.01iT - 31T^{2} \)
37 \( 1 + (5.30 + 5.30i)T + 37iT^{2} \)
41 \( 1 - 3.42iT - 41T^{2} \)
43 \( 1 + (-4.67 + 4.67i)T - 43iT^{2} \)
47 \( 1 + (6.51 - 6.51i)T - 47iT^{2} \)
53 \( 1 + (-6.18 + 6.18i)T - 53iT^{2} \)
59 \( 1 - 9.39T + 59T^{2} \)
61 \( 1 - 1.99iT - 61T^{2} \)
67 \( 1 + (0.224 + 0.224i)T + 67iT^{2} \)
71 \( 1 - 7.88T + 71T^{2} \)
73 \( 1 + (9.62 + 9.62i)T + 73iT^{2} \)
79 \( 1 + 8.59iT - 79T^{2} \)
83 \( 1 + (-6.32 - 6.32i)T + 83iT^{2} \)
89 \( 1 + 1.04T + 89T^{2} \)
97 \( 1 + (4.17 - 4.17i)T - 97iT^{2} \)
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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

−9.412303082795844974226951341180, −8.873546855445052875860480395630, −7.87326461123876198395087349835, −7.00227878886569759948652439370, −6.26810190739128832366458623236, −5.03730952656165007025943257925, −4.09215620285460308005119042591, −3.41519824582431015974148535219, −2.50599188634054749476705836606, −0.60498498775213869542892464260, 1.63645951083893913503893430650, 2.38354618937910989805320870283, 3.52278690791825744981910136531, 4.48751473685472607628827807886, 5.74227460531425578557038949099, 6.75263567657426663375155120934, 7.02332956304427050716322556751, 8.294788745739872128192579713525, 8.769745164079012185614452864686, 9.530526857631864397124541314259

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