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Properties

Label 2-1700-1700.147-c0-0-0
Degree $2$
Conductor $1700$
Sign $0.853 - 0.521i$
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  + (−0.996 + 0.0784i)2-s + (0.987 − 0.156i)4-s + (−0.987 − 0.156i)5-s + (−0.972 + 0.233i)8-s + (−0.760 + 0.649i)9-s + (0.996 + 0.0784i)10-s + (0.444 − 0.144i)13-s + (0.951 − 0.309i)16-s + (0.453 − 0.891i)17-s + (0.707 − 0.707i)18-s − 20-s + (0.951 + 0.309i)25-s + (−0.431 + 0.178i)26-s + (0.666 + 1.80i)29-s + (−0.923 + 0.382i)32-s + ⋯
L(s)  = 1  + (−0.996 + 0.0784i)2-s + (0.987 − 0.156i)4-s + (−0.987 − 0.156i)5-s + (−0.972 + 0.233i)8-s + (−0.760 + 0.649i)9-s + (0.996 + 0.0784i)10-s + (0.444 − 0.144i)13-s + (0.951 − 0.309i)16-s + (0.453 − 0.891i)17-s + (0.707 − 0.707i)18-s − 20-s + (0.951 + 0.309i)25-s + (−0.431 + 0.178i)26-s + (0.666 + 1.80i)29-s + (−0.923 + 0.382i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.853 - 0.521i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.853 - 0.521i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1700\)    =    \(2^{2} \cdot 5^{2} \cdot 17\)
Sign: $0.853 - 0.521i$
Analytic conductor: \(0.848410\)
Root analytic conductor: \(0.921092\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1700} (147, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1700,\ (\ :0),\ 0.853 - 0.521i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5588418143\)
\(L(\frac12)\) \(\approx\) \(0.5588418143\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.996 - 0.0784i)T \)
5 \( 1 + (0.987 + 0.156i)T \)
17 \( 1 + (-0.453 + 0.891i)T \)
good3 \( 1 + (0.760 - 0.649i)T^{2} \)
7 \( 1 + (-0.923 - 0.382i)T^{2} \)
11 \( 1 + (0.522 + 0.852i)T^{2} \)
13 \( 1 + (-0.444 + 0.144i)T + (0.809 - 0.587i)T^{2} \)
19 \( 1 + (0.891 - 0.453i)T^{2} \)
23 \( 1 + (-0.852 + 0.522i)T^{2} \)
29 \( 1 + (-0.666 - 1.80i)T + (-0.760 + 0.649i)T^{2} \)
31 \( 1 + (0.0784 + 0.996i)T^{2} \)
37 \( 1 + (-0.226 - 0.0638i)T + (0.852 + 0.522i)T^{2} \)
41 \( 1 + (-1.56 - 1.23i)T + (0.233 + 0.972i)T^{2} \)
43 \( 1 + (0.707 + 0.707i)T^{2} \)
47 \( 1 + (-0.309 - 0.951i)T^{2} \)
53 \( 1 + (1.01 + 0.243i)T + (0.891 + 0.453i)T^{2} \)
59 \( 1 + (0.156 - 0.987i)T^{2} \)
61 \( 1 + (-1.41 + 0.398i)T + (0.852 - 0.522i)T^{2} \)
67 \( 1 + (0.951 + 0.309i)T^{2} \)
71 \( 1 + (0.649 + 0.760i)T^{2} \)
73 \( 1 + (0.0984 - 0.831i)T + (-0.972 - 0.233i)T^{2} \)
79 \( 1 + (-0.0784 + 0.996i)T^{2} \)
83 \( 1 + (-0.453 - 0.891i)T^{2} \)
89 \( 1 + (-0.0712 - 0.139i)T + (-0.587 + 0.809i)T^{2} \)
97 \( 1 + (-1.42 - 0.657i)T + (0.649 + 0.760i)T^{2} \)
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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

−9.406728931324951987835948278363, −8.714003167785294719346158219335, −8.073979487627177811834951232829, −7.48646850885876466735922301423, −6.66902861432196667334661191163, −5.62772017522920335138554946750, −4.77389199071676007318147459860, −3.39358319281260119597218329807, −2.62724042075182796501868190859, −1.04185245266334702473969637005, 0.76218979166036468935683884055, 2.38818698831577730971186892906, 3.41898475360946198716131101417, 4.16711951501813260372940704529, 5.77254565491351866443424312169, 6.35532187709789006162917204146, 7.31533181073213775788686740261, 8.052612592380734033544348572721, 8.589254142191724018505262026097, 9.327168267297243099459266504453

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