Properties

Label 2-700-140.139-c1-0-33
Degree $2$
Conductor $700$
Sign $0.991 - 0.129i$
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  + (0.366 + 1.36i)2-s − 1.73i·3-s + (−1.73 + i)4-s + (2.36 − 0.633i)6-s + (−2 + 1.73i)7-s + (−2 − 1.99i)8-s − 3.73i·11-s + (1.73 + 2.99i)12-s + 6.46·13-s + (−3.09 − 2.09i)14-s + (1.99 − 3.46i)16-s − 0.464·17-s + 6·19-s + (2.99 + 3.46i)21-s + (5.09 − 1.36i)22-s − 5.46·23-s + ⋯
L(s)  = 1  + (0.258 + 0.965i)2-s − 0.999i·3-s + (−0.866 + 0.5i)4-s + (0.965 − 0.258i)6-s + (−0.755 + 0.654i)7-s + (−0.707 − 0.707i)8-s − 1.12i·11-s + (0.499 + 0.866i)12-s + 1.79·13-s + (−0.827 − 0.560i)14-s + (0.499 − 0.866i)16-s − 0.112·17-s + 1.37·19-s + (0.654 + 0.755i)21-s + (1.08 − 0.291i)22-s − 1.13·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.129i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.129i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.991 - 0.129i$
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: \((2,\ 700,\ (\ :1/2),\ 0.991 - 0.129i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.55669 + 0.101464i\)
\(L(\frac12)\) \(\approx\) \(1.55669 + 0.101464i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.366 - 1.36i)T \)
5 \( 1 \)
7 \( 1 + (2 - 1.73i)T \)
good3 \( 1 + 1.73iT - 3T^{2} \)
11 \( 1 + 3.73iT - 11T^{2} \)
13 \( 1 - 6.46T + 13T^{2} \)
17 \( 1 + 0.464T + 17T^{2} \)
19 \( 1 - 6T + 19T^{2} \)
23 \( 1 + 5.46T + 23T^{2} \)
29 \( 1 - 5.92T + 29T^{2} \)
31 \( 1 - 6T + 31T^{2} \)
37 \( 1 - 2.53iT - 37T^{2} \)
41 \( 1 + 3.46iT - 41T^{2} \)
43 \( 1 - 2T + 43T^{2} \)
47 \( 1 + 1.73iT - 47T^{2} \)
53 \( 1 - 2iT - 53T^{2} \)
59 \( 1 - 3.46T + 59T^{2} \)
61 \( 1 + 2.53iT - 61T^{2} \)
67 \( 1 + 3.46T + 67T^{2} \)
71 \( 1 + 0.535iT - 71T^{2} \)
73 \( 1 + 0.928T + 73T^{2} \)
79 \( 1 + 2.66iT - 79T^{2} \)
83 \( 1 + 8.53iT - 83T^{2} \)
89 \( 1 - 9.46iT - 89T^{2} \)
97 \( 1 + 7.39T + 97T^{2} \)
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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

−10.32751945077261457980470506153, −9.266690146988573226806382852345, −8.423140018597090461830975662498, −7.902994492317702195605125558852, −6.65820620006441924436404157661, −6.23376857038188056283950389357, −5.49898071924011538919163439393, −3.93006025932532165103262718221, −2.95987339637686428330878915795, −0.962026705807218063465035167858, 1.27501318462483227423971621274, 3.01975744552600554714090710833, 3.94604705228368635446209077612, 4.47835105676622924650221636379, 5.68414868100359874747664237165, 6.74613318462833870797045210224, 8.092403427031139083560524204027, 9.187030018519288153772845426188, 9.888075740293207772038036226477, 10.27277883020486951586787056791

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