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Properties

Label 2-700-140.19-c1-0-62
Degree $2$
Conductor $700$
Sign $-0.995 - 0.0992i$
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  + (−1.39 + 0.237i)2-s + (0.623 − 0.359i)3-s + (1.88 − 0.661i)4-s + (−0.783 + 0.649i)6-s + (0.956 − 2.46i)7-s + (−2.47 + 1.37i)8-s + (−1.24 + 2.14i)9-s + (−3.10 + 1.79i)11-s + (0.938 − 1.09i)12-s − 5.76·13-s + (−0.747 + 3.66i)14-s + (3.12 − 2.49i)16-s + (−3.84 − 6.65i)17-s + (1.22 − 3.29i)18-s + (−1.03 + 1.79i)19-s + ⋯
L(s)  = 1  + (−0.985 + 0.167i)2-s + (0.359 − 0.207i)3-s + (0.943 − 0.330i)4-s + (−0.319 + 0.265i)6-s + (0.361 − 0.932i)7-s + (−0.874 + 0.484i)8-s + (−0.413 + 0.716i)9-s + (−0.937 + 0.541i)11-s + (0.270 − 0.315i)12-s − 1.59·13-s + (−0.199 + 0.979i)14-s + (0.781 − 0.624i)16-s + (−0.932 − 1.61i)17-s + (0.287 − 0.775i)18-s + (−0.237 + 0.411i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 - 0.0992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.995 - 0.0992i$
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: \((2,\ 700,\ (\ :1/2),\ -0.995 - 0.0992i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.000507384 + 0.0102007i\)
\(L(\frac12)\) \(\approx\) \(0.000507384 + 0.0102007i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.39 - 0.237i)T \)
5 \( 1 \)
7 \( 1 + (-0.956 + 2.46i)T \)
good3 \( 1 + (-0.623 + 0.359i)T + (1.5 - 2.59i)T^{2} \)
11 \( 1 + (3.10 - 1.79i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 5.76T + 13T^{2} \)
17 \( 1 + (3.84 + 6.65i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.03 - 1.79i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.73 - 6.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 0.948T + 29T^{2} \)
31 \( 1 + (-0.252 - 0.437i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.99 - 1.72i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 4.24iT - 41T^{2} \)
43 \( 1 - 10.5T + 43T^{2} \)
47 \( 1 + (5.36 + 3.09i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (5.11 - 2.95i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.73 - 4.74i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.42 + 1.97i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.05 + 3.56i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 0.107iT - 71T^{2} \)
73 \( 1 + (4.67 + 8.09i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.63 + 2.10i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 11.9iT - 83T^{2} \)
89 \( 1 + (2.28 + 1.32i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 3.28T + 97T^{2} \)
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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

−9.892623007064023779231131209393, −9.256127393035693283032116428253, −8.000848042969213862398893534213, −7.53618546416423752316335662629, −7.04025933299035384634826570473, −5.49075914767290584424563980794, −4.64410064497006183473143120557, −2.79927671775504431583757007494, −1.97074817683305364393156046830, −0.00620569292794742789241351227, 2.21687868924886097585727668103, 2.83479428691862675502483574076, 4.36777919079002623346440477800, 5.78638667084421602894982200249, 6.52280681044053352264023918074, 7.84588153335055276719232276558, 8.410894233464662996280146571834, 9.087849539994333242784804646347, 9.920415167563288615219803684562, 10.75485457146427643447667301504

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