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Properties

Label 2-1700-1700.1539-c0-0-1
Degree $2$
Conductor $1700$
Sign $0.946 - 0.323i$
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  + (0.987 + 0.156i)2-s + (0.951 + 0.309i)4-s + (0.951 − 0.309i)5-s + (0.891 + 0.453i)8-s + (0.156 + 0.987i)9-s + (0.987 − 0.156i)10-s + (−1.44 − 1.04i)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 + (−1.26 − 1.26i)26-s + (−1.29 − 1.10i)29-s + (0.707 + 0.707i)32-s + ⋯
L(s)  = 1  + (0.987 + 0.156i)2-s + (0.951 + 0.309i)4-s + (0.951 − 0.309i)5-s + (0.891 + 0.453i)8-s + (0.156 + 0.987i)9-s + (0.987 − 0.156i)10-s + (−1.44 − 1.04i)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 + (−1.26 − 1.26i)26-s + (−1.29 − 1.10i)29-s + (0.707 + 0.707i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.946 - 0.323i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.946 - 0.323i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1700\)    =    \(2^{2} \cdot 5^{2} \cdot 17\)
Sign: $0.946 - 0.323i$
Analytic conductor: \(0.848410\)
Root analytic conductor: \(0.921092\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1700} (1539, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1700,\ (\ :0),\ 0.946 - 0.323i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.319802402\)
\(L(\frac12)\) \(\approx\) \(2.319802402\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.987 - 0.156i)T \)
5 \( 1 + (-0.951 + 0.309i)T \)
17 \( 1 + (0.587 - 0.809i)T \)
good3 \( 1 + (-0.156 - 0.987i)T^{2} \)
7 \( 1 + (-0.707 + 0.707i)T^{2} \)
11 \( 1 + (0.453 + 0.891i)T^{2} \)
13 \( 1 + (1.44 + 1.04i)T + (0.309 + 0.951i)T^{2} \)
19 \( 1 + (-0.587 - 0.809i)T^{2} \)
23 \( 1 + (-0.453 - 0.891i)T^{2} \)
29 \( 1 + (1.29 + 1.10i)T + (0.156 + 0.987i)T^{2} \)
31 \( 1 + (0.987 + 0.156i)T^{2} \)
37 \( 1 + (1.65 - 1.01i)T + (0.453 - 0.891i)T^{2} \)
41 \( 1 + (-0.465 + 1.93i)T + (-0.891 - 0.453i)T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + (0.809 + 0.587i)T^{2} \)
53 \( 1 + (0.809 - 0.412i)T + (0.587 - 0.809i)T^{2} \)
59 \( 1 + (0.951 - 0.309i)T^{2} \)
61 \( 1 + (-0.133 - 0.0819i)T + (0.453 + 0.891i)T^{2} \)
67 \( 1 + (-0.809 + 0.587i)T^{2} \)
71 \( 1 + (0.156 + 0.987i)T^{2} \)
73 \( 1 + (-1.26 + 0.303i)T + (0.891 - 0.453i)T^{2} \)
79 \( 1 + (0.987 - 0.156i)T^{2} \)
83 \( 1 + (0.587 + 0.809i)T^{2} \)
89 \( 1 + (-1.16 - 1.59i)T + (-0.309 + 0.951i)T^{2} \)
97 \( 1 + (-0.303 + 0.355i)T + (-0.156 - 0.987i)T^{2} \)
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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

−9.791468625858565042527483895034, −8.645311989240697177368200193375, −7.78614354197428739283487728870, −7.12552625171929531281277754476, −6.12507174319746145319322138728, −5.31617584835635326620471041686, −4.92555025878027446790384564310, −3.79554916562065850481402846154, −2.48430602804289884098991289874, −1.93923873975810357965920245070, 1.68544135996880037208631605018, 2.57101032518208175489836952859, 3.53973592794457198542090430071, 4.63366303976047554891023254406, 5.29999117223819565623159357332, 6.28163213715379768272409800448, 6.91136464068685741855609004363, 7.42283227478195445058330672130, 9.125963592084453929352838440860, 9.462724986996215488480341750226

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