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Properties

Label 2-1700-1700.1039-c0-0-1
Degree $2$
Conductor $1700$
Sign $0.234 + 0.972i$
Analytic cond. $0.848410$
Root an. cond. $0.921092$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.156 − 0.987i)2-s + (−0.951 − 0.309i)4-s + (−0.951 + 0.309i)5-s + (−0.453 + 0.891i)8-s + (0.987 − 0.156i)9-s + (0.156 + 0.987i)10-s + (0.734 + 0.533i)13-s + (0.809 + 0.587i)16-s + (0.587 − 0.809i)17-s − i·18-s + 20-s + (0.809 − 0.587i)25-s + (0.642 − 0.642i)26-s + (−0.465 + 0.0366i)29-s + (0.707 − 0.707i)32-s + ⋯
L(s)  = 1  + (0.156 − 0.987i)2-s + (−0.951 − 0.309i)4-s + (−0.951 + 0.309i)5-s + (−0.453 + 0.891i)8-s + (0.987 − 0.156i)9-s + (0.156 + 0.987i)10-s + (0.734 + 0.533i)13-s + (0.809 + 0.587i)16-s + (0.587 − 0.809i)17-s − i·18-s + 20-s + (0.809 − 0.587i)25-s + (0.642 − 0.642i)26-s + (−0.465 + 0.0366i)29-s + (0.707 − 0.707i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.234 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.234 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1700\)    =    \(2^{2} \cdot 5^{2} \cdot 17\)
Sign: $0.234 + 0.972i$
Analytic conductor: \(0.848410\)
Root analytic conductor: \(0.921092\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1700} (1039, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1700,\ (\ :0),\ 0.234 + 0.972i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.023560282\)
\(L(\frac12)\) \(\approx\) \(1.023560282\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.156 + 0.987i)T \)
5 \( 1 + (0.951 - 0.309i)T \)
17 \( 1 + (-0.587 + 0.809i)T \)
good3 \( 1 + (-0.987 + 0.156i)T^{2} \)
7 \( 1 + (-0.707 - 0.707i)T^{2} \)
11 \( 1 + (-0.891 + 0.453i)T^{2} \)
13 \( 1 + (-0.734 - 0.533i)T + (0.309 + 0.951i)T^{2} \)
19 \( 1 + (0.587 + 0.809i)T^{2} \)
23 \( 1 + (0.891 - 0.453i)T^{2} \)
29 \( 1 + (0.465 - 0.0366i)T + (0.987 - 0.156i)T^{2} \)
31 \( 1 + (0.156 - 0.987i)T^{2} \)
37 \( 1 + (-0.243 + 1.01i)T + (-0.891 - 0.453i)T^{2} \)
41 \( 1 + (-1.29 + 0.794i)T + (0.453 - 0.891i)T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + (0.809 + 0.587i)T^{2} \)
53 \( 1 + (0.809 + 1.58i)T + (-0.587 + 0.809i)T^{2} \)
59 \( 1 + (-0.951 + 0.309i)T^{2} \)
61 \( 1 + (-0.303 - 1.26i)T + (-0.891 + 0.453i)T^{2} \)
67 \( 1 + (-0.809 + 0.587i)T^{2} \)
71 \( 1 + (0.987 - 0.156i)T^{2} \)
73 \( 1 + (0.0819 - 0.133i)T + (-0.453 - 0.891i)T^{2} \)
79 \( 1 + (0.156 + 0.987i)T^{2} \)
83 \( 1 + (-0.587 - 0.809i)T^{2} \)
89 \( 1 + (0.183 + 0.253i)T + (-0.309 + 0.951i)T^{2} \)
97 \( 1 + (-0.133 - 1.70i)T + (-0.987 + 0.156i)T^{2} \)
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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

−9.435044044381158409625149049611, −8.811427925520874066059853648033, −7.78277860176848294191916754841, −7.14867374085626483080036910030, −6.04752655997511602911779991338, −4.94052289336313031256562978626, −4.06054304061874433003140868262, −3.56401379160342003392267714629, −2.37442136467227705873612271158, −1.01895331307235346970436082094, 1.17841441402173499584502073447, 3.26463874177786480926072335759, 4.04929793684059510875875924749, 4.72572554698676176657010841783, 5.72483332222134281141714400923, 6.53748337422486268295506711457, 7.50102899865609930574760627708, 7.925954543502378092545270462034, 8.632590242598946440853987628774, 9.508886217435154291197327270271

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