Properties

Label 2-1400-35.13-c1-0-22
Degree $2$
Conductor $1400$
Sign $0.793 + 0.608i$
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  + (0.730 − 0.730i)3-s + (−2.41 − 1.08i)7-s + 1.93i·9-s + 5.95·11-s + (0.921 − 0.921i)13-s + (−2.02 − 2.02i)17-s + 3.29·19-s + (−2.55 + 0.970i)21-s + (−0.0544 − 0.0544i)23-s + (3.60 + 3.60i)27-s − 3.78i·29-s + 4.88i·31-s + (4.35 − 4.35i)33-s + (3.20 − 3.20i)37-s − 1.34i·39-s + ⋯
L(s)  = 1  + (0.421 − 0.421i)3-s + (−0.912 − 0.409i)7-s + 0.644i·9-s + 1.79·11-s + (0.255 − 0.255i)13-s + (−0.491 − 0.491i)17-s + 0.755·19-s + (−0.557 + 0.211i)21-s + (−0.0113 − 0.0113i)23-s + (0.693 + 0.693i)27-s − 0.703i·29-s + 0.876i·31-s + (0.757 − 0.757i)33-s + (0.527 − 0.527i)37-s − 0.215i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.793 + 0.608i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.793 + 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $0.793 + 0.608i$
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: \((2,\ 1400,\ (\ :1/2),\ 0.793 + 0.608i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.956168260\)
\(L(\frac12)\) \(\approx\) \(1.956168260\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.41 + 1.08i)T \)
good3 \( 1 + (-0.730 + 0.730i)T - 3iT^{2} \)
11 \( 1 - 5.95T + 11T^{2} \)
13 \( 1 + (-0.921 + 0.921i)T - 13iT^{2} \)
17 \( 1 + (2.02 + 2.02i)T + 17iT^{2} \)
19 \( 1 - 3.29T + 19T^{2} \)
23 \( 1 + (0.0544 + 0.0544i)T + 23iT^{2} \)
29 \( 1 + 3.78iT - 29T^{2} \)
31 \( 1 - 4.88iT - 31T^{2} \)
37 \( 1 + (-3.20 + 3.20i)T - 37iT^{2} \)
41 \( 1 + 10.6iT - 41T^{2} \)
43 \( 1 + (-1.60 - 1.60i)T + 43iT^{2} \)
47 \( 1 + (2.64 + 2.64i)T + 47iT^{2} \)
53 \( 1 + (-9.16 - 9.16i)T + 53iT^{2} \)
59 \( 1 - 13.3T + 59T^{2} \)
61 \( 1 + 11.2iT - 61T^{2} \)
67 \( 1 + (-6.43 + 6.43i)T - 67iT^{2} \)
71 \( 1 + 8.51T + 71T^{2} \)
73 \( 1 + (-8.66 + 8.66i)T - 73iT^{2} \)
79 \( 1 + 1.76iT - 79T^{2} \)
83 \( 1 + (-2.36 + 2.36i)T - 83iT^{2} \)
89 \( 1 + 9.73T + 89T^{2} \)
97 \( 1 + (-2.05 - 2.05i)T + 97iT^{2} \)
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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

−9.296568756393207773650479864263, −8.844837558041501603248230682951, −7.76245690019604182897310674752, −7.00748980355907762720662871600, −6.44836746128814599220679314465, −5.37233670199617090012330043187, −4.15409683891347768187413901106, −3.39875704078595773804869081104, −2.24894348441481092678335455417, −0.947279912695812496546305480354, 1.17321116014902063944422924275, 2.71392268404374791878176453128, 3.71645707885940442895735577237, 4.18385670615665042347978410415, 5.64003462200874379227371860383, 6.51466443509827728575648058027, 6.90750358524456075122329302715, 8.374353327035540902195431013788, 8.957349135395921951340642697475, 9.605003594248254809236728978433

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