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Properties

Label 2-1700-1700.1039-c0-0-0
Degree $2$
Conductor $1700$
Sign $-0.523 - 0.852i$
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 + (−0.453 + 0.891i)5-s + (0.453 − 0.891i)8-s + (0.987 − 0.156i)9-s + (−0.809 − 0.587i)10-s + (1.44 + 1.04i)13-s + (0.809 + 0.587i)16-s + (−0.951 − 0.309i)17-s + i·18-s + (0.707 − 0.707i)20-s + (−0.587 − 0.809i)25-s + (−1.26 + 1.26i)26-s + (1.70 − 0.133i)29-s + (−0.707 + 0.707i)32-s + ⋯
L(s)  = 1  + (−0.156 + 0.987i)2-s + (−0.951 − 0.309i)4-s + (−0.453 + 0.891i)5-s + (0.453 − 0.891i)8-s + (0.987 − 0.156i)9-s + (−0.809 − 0.587i)10-s + (1.44 + 1.04i)13-s + (0.809 + 0.587i)16-s + (−0.951 − 0.309i)17-s + i·18-s + (0.707 − 0.707i)20-s + (−0.587 − 0.809i)25-s + (−1.26 + 1.26i)26-s + (1.70 − 0.133i)29-s + (−0.707 + 0.707i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.523 - 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.523 - 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1700\)    =    \(2^{2} \cdot 5^{2} \cdot 17\)
Sign: $-0.523 - 0.852i$
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: \((2,\ 1700,\ (\ :0),\ -0.523 - 0.852i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9420840750\)
\(L(\frac12)\) \(\approx\) \(0.9420840750\)
\(L(1)\) 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 \( 1 + (0.453 - 0.891i)T \)
17 \( 1 + (0.951 + 0.309i)T \)
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 \( 1 + (-1.44 - 1.04i)T + (0.309 + 0.951i)T^{2} \)
19 \( 1 + (0.587 + 0.809i)T^{2} \)
23 \( 1 + (0.891 - 0.453i)T^{2} \)
29 \( 1 + (-1.70 + 0.133i)T + (0.987 - 0.156i)T^{2} \)
31 \( 1 + (0.156 - 0.987i)T^{2} \)
37 \( 1 + (0.453 - 1.89i)T + (-0.891 - 0.453i)T^{2} \)
41 \( 1 + (1.70 - 1.04i)T + (0.453 - 0.891i)T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + (0.809 + 0.587i)T^{2} \)
53 \( 1 + (0.412 + 0.809i)T + (-0.587 + 0.809i)T^{2} \)
59 \( 1 + (-0.951 + 0.309i)T^{2} \)
61 \( 1 + (0.0366 + 0.152i)T + (-0.891 + 0.453i)T^{2} \)
67 \( 1 + (-0.809 + 0.587i)T^{2} \)
71 \( 1 + (0.987 - 0.156i)T^{2} \)
73 \( 1 + (0.678 - 1.10i)T + (-0.453 - 0.891i)T^{2} \)
79 \( 1 + (0.156 + 0.987i)T^{2} \)
83 \( 1 + (-0.587 - 0.809i)T^{2} \)
89 \( 1 + (1.16 + 1.59i)T + (-0.309 + 0.951i)T^{2} \)
97 \( 1 + (-0.0366 - 0.465i)T + (-0.987 + 0.156i)T^{2} \)
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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

−9.807880620417980475783348980795, −8.698765769568213336406919366593, −8.280986124891590766203834812606, −7.17179391331245557716079548066, −6.62175549601872841130729456143, −6.25993234492816318353395212319, −4.74654193972578529090032799404, −4.19392139375904041299922561684, −3.20939115047520329056420011733, −1.47864970698250414900136419985, 0.900272728950032762836475207389, 1.95305115320687031326304457509, 3.40471752079174629140720323449, 4.09869553057789175515415369519, 4.87166795361862373308576041027, 5.78528452879629299737992229050, 7.05847276561196231759895668392, 8.047016390043585800842260299309, 8.623587518775309783563091014761, 9.156511917381591949617321648406

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