Properties

Label 2-700-5.4-c5-0-20
Degree $2$
Conductor $700$
Sign $0.894 - 0.447i$
Analytic cond. $112.268$
Root an. cond. $10.5956$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.57i·3-s − 49i·7-s + 240.·9-s + 322.·11-s + 657. i·13-s − 665. i·17-s + 2.21e3·19-s + 77.3·21-s + 1.58e3i·23-s + 763. i·27-s − 6.38e3·29-s + 7.71e3·31-s + 509. i·33-s − 3.43e3i·37-s − 1.03e3·39-s + ⋯
L(s)  = 1  + 0.101i·3-s − 0.377i·7-s + 0.989·9-s + 0.804·11-s + 1.07i·13-s − 0.558i·17-s + 1.40·19-s + 0.0382·21-s + 0.625i·23-s + 0.201i·27-s − 1.41·29-s + 1.44·31-s + 0.0814i·33-s − 0.412i·37-s − 0.109·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.894 - 0.447i$
Analytic conductor: \(112.268\)
Root analytic conductor: \(10.5956\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{700} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 700,\ (\ :5/2),\ 0.894 - 0.447i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.724093502\)
\(L(\frac12)\) \(\approx\) \(2.724093502\)
\(L(\frac{7}{2})\) 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 \( 1 + 49iT \)
good3 \( 1 - 1.57iT - 243T^{2} \)
11 \( 1 - 322.T + 1.61e5T^{2} \)
13 \( 1 - 657. iT - 3.71e5T^{2} \)
17 \( 1 + 665. iT - 1.41e6T^{2} \)
19 \( 1 - 2.21e3T + 2.47e6T^{2} \)
23 \( 1 - 1.58e3iT - 6.43e6T^{2} \)
29 \( 1 + 6.38e3T + 2.05e7T^{2} \)
31 \( 1 - 7.71e3T + 2.86e7T^{2} \)
37 \( 1 + 3.43e3iT - 6.93e7T^{2} \)
41 \( 1 + 6.53e3T + 1.15e8T^{2} \)
43 \( 1 - 1.66e4iT - 1.47e8T^{2} \)
47 \( 1 + 1.25e4iT - 2.29e8T^{2} \)
53 \( 1 - 4.24e3iT - 4.18e8T^{2} \)
59 \( 1 + 2.61e4T + 7.14e8T^{2} \)
61 \( 1 - 3.26e4T + 8.44e8T^{2} \)
67 \( 1 - 6.15e3iT - 1.35e9T^{2} \)
71 \( 1 + 3.38e4T + 1.80e9T^{2} \)
73 \( 1 + 8.21e4iT - 2.07e9T^{2} \)
79 \( 1 - 3.52e4T + 3.07e9T^{2} \)
83 \( 1 - 1.75e4iT - 3.93e9T^{2} \)
89 \( 1 - 7.28e4T + 5.58e9T^{2} \)
97 \( 1 - 1.66e5iT - 8.58e9T^{2} \)
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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

−9.499340298393855367704736020803, −9.291439422497148848854007282137, −7.82857541631037873905344433437, −7.13474328882530128501090294950, −6.37911322901830627403363950449, −5.09245439138985207923524462621, −4.21640115367422201315069247247, −3.35915323752538163270186320249, −1.81570999956989199154071814285, −0.935261937219353378877836082902, 0.72911885323636821140721839518, 1.71484268156701881966570971454, 3.05798328581133508057363590606, 4.04472661761417354231616865350, 5.14171379280802124512533229010, 6.07464537048818814302180258331, 7.03271910214364520705850199251, 7.84933666250167715147436787004, 8.758352385043294761698034474623, 9.736142875789727026441194762662

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