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Properties

Label 2-700-140.19-c1-0-60
Degree $2$
Conductor $700$
Sign $-0.587 + 0.809i$
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  + (0.468 − 1.33i)2-s + (2.22 − 1.28i)3-s + (−1.56 − 1.25i)4-s + (−0.670 − 3.57i)6-s + (2.62 − 0.304i)7-s + (−2.40 + 1.49i)8-s + (1.80 − 3.12i)9-s + (5.44 − 3.14i)11-s + (−5.08 − 0.780i)12-s − 3.00·13-s + (0.826 − 3.64i)14-s + (0.868 + 3.90i)16-s + (0.539 + 0.935i)17-s + (−3.32 − 3.87i)18-s + (−3.18 + 5.51i)19-s + ⋯
L(s)  = 1  + (0.331 − 0.943i)2-s + (1.28 − 0.742i)3-s + (−0.780 − 0.625i)4-s + (−0.273 − 1.45i)6-s + (0.993 − 0.115i)7-s + (−0.848 + 0.528i)8-s + (0.601 − 1.04i)9-s + (1.64 − 0.947i)11-s + (−1.46 − 0.225i)12-s − 0.834·13-s + (0.220 − 0.975i)14-s + (0.217 + 0.976i)16-s + (0.130 + 0.226i)17-s + (−0.783 − 0.912i)18-s + (−0.730 + 1.26i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.587 + 0.809i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.587 + 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.587 + 0.809i$
Analytic conductor: \(5.58952\)
Root analytic conductor: \(2.36421\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{700} (299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 700,\ (\ :1/2),\ -0.587 + 0.809i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.24494 - 2.44055i\)
\(L(\frac12)\) \(\approx\) \(1.24494 - 2.44055i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.468 + 1.33i)T \)
5 \( 1 \)
7 \( 1 + (-2.62 + 0.304i)T \)
good3 \( 1 + (-2.22 + 1.28i)T + (1.5 - 2.59i)T^{2} \)
11 \( 1 + (-5.44 + 3.14i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 3.00T + 13T^{2} \)
17 \( 1 + (-0.539 - 0.935i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.18 - 5.51i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.60 - 2.77i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 0.512T + 29T^{2} \)
31 \( 1 + (2.55 + 4.41i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.56 + 2.63i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 7.61iT - 41T^{2} \)
43 \( 1 + 0.683T + 43T^{2} \)
47 \( 1 + (9.71 + 5.61i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.93 + 1.11i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.62 + 2.80i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.28 - 1.31i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.78 - 6.55i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 13.2iT - 71T^{2} \)
73 \( 1 + (1.26 + 2.18i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-9.18 - 5.30i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 1.37iT - 83T^{2} \)
89 \( 1 + (-7.79 - 4.49i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 7.56T + 97T^{2} \)
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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

−10.07452519877778744655842228564, −9.236010261535486628402077859348, −8.458034716963910046386980435238, −7.916212370432252767280941609846, −6.65046613715116681437815878962, −5.51355517532483452304558581674, −4.11128793596180521003413063532, −3.45926288660892024592511613463, −2.10534499860074573345518368233, −1.37989407784403224110443750697, 2.14870734166568009830161433479, 3.50614875210145844927614706912, 4.49674622623759825180870807672, 4.90477384026585883873271019723, 6.54672453257114098014647800946, 7.32306495236009092281539578409, 8.223294455157577274571373950418, 9.055978843544071966684850583832, 9.345095249468514451903826669020, 10.45859181474759649176162752863

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