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Properties

Label 2-700-5.4-c3-0-4
Degree $2$
Conductor $700$
Sign $-0.894 + 0.447i$
Analytic cond. $41.3013$
Root an. cond. $6.42661$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 8i·3-s − 7i·7-s − 37·9-s + 28·11-s + 82i·13-s + 46i·17-s − 8·19-s + 56·21-s − 128i·23-s − 80i·27-s − 174·29-s − 152·31-s + 224i·33-s + 290i·37-s − 656·39-s + ⋯
L(s)  = 1  + 1.53i·3-s − 0.377i·7-s − 1.37·9-s + 0.767·11-s + 1.74i·13-s + 0.656i·17-s − 0.0965·19-s + 0.581·21-s − 1.16i·23-s − 0.570i·27-s − 1.11·29-s − 0.880·31-s + 1.18i·33-s + 1.28i·37-s − 2.69·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.894 + 0.447i$
Analytic conductor: \(41.3013\)
Root analytic conductor: \(6.42661\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{700} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 700,\ (\ :3/2),\ -0.894 + 0.447i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.103718828\)
\(L(\frac12)\) \(\approx\) \(1.103718828\)
\(L(\frac{5}{2})\) 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 \( 1 + 7iT \)
good3 \( 1 - 8iT - 27T^{2} \)
11 \( 1 - 28T + 1.33e3T^{2} \)
13 \( 1 - 82iT - 2.19e3T^{2} \)
17 \( 1 - 46iT - 4.91e3T^{2} \)
19 \( 1 + 8T + 6.85e3T^{2} \)
23 \( 1 + 128iT - 1.21e4T^{2} \)
29 \( 1 + 174T + 2.43e4T^{2} \)
31 \( 1 + 152T + 2.97e4T^{2} \)
37 \( 1 - 290iT - 5.06e4T^{2} \)
41 \( 1 - 50T + 6.89e4T^{2} \)
43 \( 1 - 396iT - 7.95e4T^{2} \)
47 \( 1 - 296iT - 1.03e5T^{2} \)
53 \( 1 + 570iT - 1.48e5T^{2} \)
59 \( 1 - 272T + 2.05e5T^{2} \)
61 \( 1 + 662T + 2.26e5T^{2} \)
67 \( 1 + 876iT - 3.00e5T^{2} \)
71 \( 1 + 880T + 3.57e5T^{2} \)
73 \( 1 + 638iT - 3.89e5T^{2} \)
79 \( 1 - 600T + 4.93e5T^{2} \)
83 \( 1 - 624iT - 5.71e5T^{2} \)
89 \( 1 + 698T + 7.04e5T^{2} \)
97 \( 1 + 754iT - 9.12e5T^{2} \)
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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

−10.55238199537600222012803758495, −9.499454328805588279025131180738, −9.228259159953760951262975320245, −8.214284632426944496340882235301, −6.87544597511449857560571261948, −6.07221787033952056893091077214, −4.70028353381260808266200760309, −4.23532740230581283573488280724, −3.35776654549675455023011525330, −1.72200907760468139473499418965, 0.30524584098653598233976479280, 1.42815544350459708405149936829, 2.53256220596016406188806003134, 3.67382707712152931985444751142, 5.46654608291612254616942162197, 5.90827927603145375737447094643, 7.24808176302326686493362470518, 7.47864079344163137652058518185, 8.572871489071158700617674233676, 9.348723809694728938475439172381

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