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Properties

Label 2-1400-35.13-c1-0-13
Degree $2$
Conductor $1400$
Sign $0.0631 - 0.998i$
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  + (−2.09 + 2.09i)3-s + (2.59 + 0.510i)7-s − 5.75i·9-s − 3.94·11-s + (1.69 − 1.69i)13-s + (2.66 + 2.66i)17-s + 5.36·19-s + (−6.50 + 4.36i)21-s + (3.55 + 3.55i)23-s + (5.77 + 5.77i)27-s − 7.24i·29-s − 0.174i·31-s + (8.26 − 8.26i)33-s + (4.85 − 4.85i)37-s + 7.10i·39-s + ⋯
L(s)  = 1  + (−1.20 + 1.20i)3-s + (0.981 + 0.193i)7-s − 1.91i·9-s − 1.19·11-s + (0.470 − 0.470i)13-s + (0.647 + 0.647i)17-s + 1.22·19-s + (−1.41 + 0.952i)21-s + (0.741 + 0.741i)23-s + (1.11 + 1.11i)27-s − 1.34i·29-s − 0.0312i·31-s + (1.43 − 1.43i)33-s + (0.798 − 0.798i)37-s + 1.13i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0631 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0631 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $0.0631 - 0.998i$
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: \((2,\ 1400,\ (\ :1/2),\ 0.0631 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.169814103\)
\(L(\frac12)\) \(\approx\) \(1.169814103\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.59 - 0.510i)T \)
good3 \( 1 + (2.09 - 2.09i)T - 3iT^{2} \)
11 \( 1 + 3.94T + 11T^{2} \)
13 \( 1 + (-1.69 + 1.69i)T - 13iT^{2} \)
17 \( 1 + (-2.66 - 2.66i)T + 17iT^{2} \)
19 \( 1 - 5.36T + 19T^{2} \)
23 \( 1 + (-3.55 - 3.55i)T + 23iT^{2} \)
29 \( 1 + 7.24iT - 29T^{2} \)
31 \( 1 + 0.174iT - 31T^{2} \)
37 \( 1 + (-4.85 + 4.85i)T - 37iT^{2} \)
41 \( 1 + 0.732iT - 41T^{2} \)
43 \( 1 + (-8.20 - 8.20i)T + 43iT^{2} \)
47 \( 1 + (-2.31 - 2.31i)T + 47iT^{2} \)
53 \( 1 + (4.53 + 4.53i)T + 53iT^{2} \)
59 \( 1 + 13.1T + 59T^{2} \)
61 \( 1 - 13.3iT - 61T^{2} \)
67 \( 1 + (6.55 - 6.55i)T - 67iT^{2} \)
71 \( 1 - 16.3T + 71T^{2} \)
73 \( 1 + (7.02 - 7.02i)T - 73iT^{2} \)
79 \( 1 + 6.63iT - 79T^{2} \)
83 \( 1 + (10.4 - 10.4i)T - 83iT^{2} \)
89 \( 1 - 3.43T + 89T^{2} \)
97 \( 1 + (1.40 + 1.40i)T + 97iT^{2} \)
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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

−9.901371765481420551616159267043, −9.229261104248480582279680872798, −8.048995114082528211203420255040, −7.49702112705716742737117686160, −5.88923954846910948620471319453, −5.63708466008710730895304966944, −4.83675522690126408788490142224, −4.02442525316880764963233492524, −2.85965759622582067216906392735, −1.02833751853881732165471022309, 0.74408642969350237436735617670, 1.70412576497788478605122165695, 2.99303520570132795884511417554, 4.75290366008183342933394313345, 5.23064927542848185622696185959, 6.03301483689573628903900147528, 7.07516337161270755943973159113, 7.53928934982824703375191035115, 8.234500932607730446544732321643, 9.350818614323398030424060002609

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