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Properties

Label 2-1700-1700.1419-c0-0-1
Degree $2$
Conductor $1700$
Sign $-0.174 + 0.984i$
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  + (0.453 − 0.891i)2-s + (−0.587 − 0.809i)4-s + (0.987 − 0.156i)5-s + (−0.987 + 0.156i)8-s + (0.891 − 0.453i)9-s + (0.309 − 0.951i)10-s + (−0.0966 + 0.297i)13-s + (−0.309 + 0.951i)16-s + (−0.587 − 0.809i)17-s − 1.00i·18-s + (−0.707 − 0.707i)20-s + (0.951 − 0.309i)25-s + (0.221 + 0.221i)26-s + (−0.152 + 0.0366i)29-s + (0.707 + 0.707i)32-s + ⋯
L(s)  = 1  + (0.453 − 0.891i)2-s + (−0.587 − 0.809i)4-s + (0.987 − 0.156i)5-s + (−0.987 + 0.156i)8-s + (0.891 − 0.453i)9-s + (0.309 − 0.951i)10-s + (−0.0966 + 0.297i)13-s + (−0.309 + 0.951i)16-s + (−0.587 − 0.809i)17-s − 1.00i·18-s + (−0.707 − 0.707i)20-s + (0.951 − 0.309i)25-s + (0.221 + 0.221i)26-s + (−0.152 + 0.0366i)29-s + (0.707 + 0.707i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.174 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.174 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1700\)    =    \(2^{2} \cdot 5^{2} \cdot 17\)
Sign: $-0.174 + 0.984i$
Analytic conductor: \(0.848410\)
Root analytic conductor: \(0.921092\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1700} (1419, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1700,\ (\ :0),\ -0.174 + 0.984i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.593242742\)
\(L(\frac12)\) \(\approx\) \(1.593242742\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.453 + 0.891i)T \)
5 \( 1 + (-0.987 + 0.156i)T \)
17 \( 1 + (0.587 + 0.809i)T \)
good3 \( 1 + (-0.891 + 0.453i)T^{2} \)
7 \( 1 + (0.707 - 0.707i)T^{2} \)
11 \( 1 + (-0.156 + 0.987i)T^{2} \)
13 \( 1 + (0.0966 - 0.297i)T + (-0.809 - 0.587i)T^{2} \)
19 \( 1 + (-0.951 + 0.309i)T^{2} \)
23 \( 1 + (0.156 - 0.987i)T^{2} \)
29 \( 1 + (0.152 - 0.0366i)T + (0.891 - 0.453i)T^{2} \)
31 \( 1 + (-0.453 + 0.891i)T^{2} \)
37 \( 1 + (-0.987 + 1.15i)T + (-0.156 - 0.987i)T^{2} \)
41 \( 1 + (-0.152 - 1.93i)T + (-0.987 + 0.156i)T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + (-0.309 + 0.951i)T^{2} \)
53 \( 1 + (1.95 + 0.309i)T + (0.951 + 0.309i)T^{2} \)
59 \( 1 + (-0.587 + 0.809i)T^{2} \)
61 \( 1 + (0.303 + 0.355i)T + (-0.156 + 0.987i)T^{2} \)
67 \( 1 + (0.309 + 0.951i)T^{2} \)
71 \( 1 + (0.891 - 0.453i)T^{2} \)
73 \( 1 + (1.70 + 0.133i)T + (0.987 + 0.156i)T^{2} \)
79 \( 1 + (-0.453 - 0.891i)T^{2} \)
83 \( 1 + (0.951 - 0.309i)T^{2} \)
89 \( 1 + (-1.69 + 0.550i)T + (0.809 - 0.587i)T^{2} \)
97 \( 1 + (-0.303 - 1.26i)T + (-0.891 + 0.453i)T^{2} \)
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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

−9.423176910821703690616571344825, −9.059433605844372911409111699317, −7.74273397272427761140390819277, −6.58128023198851445401618098707, −6.06952841755594256055726173767, −4.91690457084515393840547557202, −4.42363164694028242177701296078, −3.20776033460349958947179185278, −2.21845851579484101191622725266, −1.22551918619632178116218428808, 1.79103417969873406581896297135, 2.99056750022948203419878567197, 4.17427498575721561342830314743, 4.93063924446196686762536207783, 5.80102841742334238504343603352, 6.48986184720219241804027051681, 7.21995757392721520004561431519, 8.039574310778653961293230283641, 8.863968407857058284230888777018, 9.665295013319071934665521676318

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