Properties

Label 2-1400-280.83-c0-0-1
Degree $2$
Conductor $1400$
Sign $0.437 - 0.899i$
Analytic cond. $0.698691$
Root an. cond. $0.835877$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.965 + 0.258i)2-s + (0.866 − 0.499i)4-s + (0.707 + 0.707i)7-s + (−0.707 + 0.707i)8-s + i·9-s + 11-s + (−0.866 − 0.500i)14-s + (0.500 − 0.866i)16-s + (−0.258 − 0.965i)18-s + (−0.965 + 0.258i)22-s + (−0.707 + 0.707i)23-s + (0.965 + 0.258i)28-s − 1.73·29-s + (−0.258 + 0.965i)32-s + (0.499 + 0.866i)36-s + (−0.707 − 0.707i)37-s + ⋯
L(s)  = 1  + (−0.965 + 0.258i)2-s + (0.866 − 0.499i)4-s + (0.707 + 0.707i)7-s + (−0.707 + 0.707i)8-s + i·9-s + 11-s + (−0.866 − 0.500i)14-s + (0.500 − 0.866i)16-s + (−0.258 − 0.965i)18-s + (−0.965 + 0.258i)22-s + (−0.707 + 0.707i)23-s + (0.965 + 0.258i)28-s − 1.73·29-s + (−0.258 + 0.965i)32-s + (0.499 + 0.866i)36-s + (−0.707 − 0.707i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.437 - 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.437 - 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $0.437 - 0.899i$
Analytic conductor: \(0.698691\)
Root analytic conductor: \(0.835877\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (643, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1400,\ (\ :0),\ 0.437 - 0.899i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7807588210\)
\(L(\frac12)\) \(\approx\) \(0.7807588210\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 - 0.258i)T \)
5 \( 1 \)
7 \( 1 + (-0.707 - 0.707i)T \)
good3 \( 1 - iT^{2} \)
11 \( 1 - T + T^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
29 \( 1 + 1.73T + T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + (-1.22 - 1.22i)T + iT^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + (-1.41 + 1.41i)T - iT^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + (-1.22 + 1.22i)T - iT^{2} \)
71 \( 1 - 1.73iT - T^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 - 1.73T + T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + iT^{2} \)
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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

−9.635754429380281529415529231330, −9.132964695617909487487879674066, −8.245826346231808969877528091148, −7.70384612717524716226263793053, −6.84472893533248034604240489843, −5.77398706145538517167772085548, −5.22699470549891823701584670596, −3.88068439778240306830719214553, −2.35406280460612877117047022117, −1.60644010234487088063850946054, 0.965022928397210697938829861752, 2.09774462166278239631058458330, 3.59180547247568932918178409059, 4.14826720011724330530796871942, 5.69972148102191602228657936648, 6.64900340183406250217496340989, 7.24954797013697223163412831513, 8.115897785799339333181986945100, 8.966381993999490312474479670923, 9.464295734647794446926253130682

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