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Properties

Label 2-700-140.139-c1-0-21
Degree $2$
Conductor $700$
Sign $0.254 - 0.967i$
Analytic cond. $5.58952$
Root an. cond. $2.36421$
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  + (1.17 − 0.780i)2-s + 3.02i·3-s + (0.780 − 1.84i)4-s + (2.35 + 3.56i)6-s + (1.51 + 2.17i)7-s + (−0.516 − 2.78i)8-s − 6.12·9-s + 4.71i·11-s + (5.56 + 2.35i)12-s + 2·13-s + (3.47 + 1.38i)14-s + (−2.78 − 2.87i)16-s − 1.12·17-s + (−7.22 + 4.78i)18-s − 4.71·19-s + ⋯
L(s)  = 1  + (0.833 − 0.552i)2-s + 1.74i·3-s + (0.390 − 0.920i)4-s + (0.962 + 1.45i)6-s + (0.570 + 0.821i)7-s + (−0.182 − 0.983i)8-s − 2.04·9-s + 1.42i·11-s + (1.60 + 0.680i)12-s + 0.554·13-s + (0.929 + 0.369i)14-s + (−0.695 − 0.718i)16-s − 0.272·17-s + (−1.70 + 1.12i)18-s − 1.08·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.254 - 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.254 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.254 - 0.967i$
Analytic conductor: \(5.58952\)
Root analytic conductor: \(2.36421\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{700} (699, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 700,\ (\ :1/2),\ 0.254 - 0.967i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.90133 + 1.46589i\)
\(L(\frac12)\) \(\approx\) \(1.90133 + 1.46589i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.17 + 0.780i)T \)
5 \( 1 \)
7 \( 1 + (-1.51 - 2.17i)T \)
good3 \( 1 - 3.02iT - 3T^{2} \)
11 \( 1 - 4.71iT - 11T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + 1.12T + 17T^{2} \)
19 \( 1 + 4.71T + 19T^{2} \)
23 \( 1 - 6.41T + 23T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 - 3.39T + 31T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 + 1.12iT - 41T^{2} \)
43 \( 1 + 0.371T + 43T^{2} \)
47 \( 1 + 5.08iT - 47T^{2} \)
53 \( 1 + 2iT - 53T^{2} \)
59 \( 1 + 2.06T + 59T^{2} \)
61 \( 1 - 2iT - 61T^{2} \)
67 \( 1 - 3.76T + 67T^{2} \)
71 \( 1 + 7.36iT - 71T^{2} \)
73 \( 1 - 15.3T + 73T^{2} \)
79 \( 1 + 1.32iT - 79T^{2} \)
83 \( 1 - 3.02iT - 83T^{2} \)
89 \( 1 + 12iT - 89T^{2} \)
97 \( 1 - 1.12T + 97T^{2} \)
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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

−10.71422866737412738376309655344, −9.948713795610457057990979081948, −9.220941313311251002208921485343, −8.439890140502255979037946123002, −6.78219461248969751600157980924, −5.68415399812606203226305334044, −4.77314271689306400696595654204, −4.42204456642919698825624553445, −3.22636520657290446108436773181, −2.10753139775822160725626162976, 1.02312127413407299657382377566, 2.48673604985681579953696546758, 3.64514598458050551229320180007, 4.99560749065875854527270215773, 6.15012622062968108897932167366, 6.60308418939363986556367731998, 7.49917147284461031517614619561, 8.254878591586994591784247594810, 8.744204932384728545057118905997, 10.95329913820456743271094163169

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