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Properties

Label 2-1700-340.259-c0-0-0
Degree $2$
Conductor $1700$
Sign $0.429 - 0.902i$
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  − 2-s + 4-s − 8-s + i·9-s + 16-s + i·17-s − i·18-s + (−1 + i)29-s − 32-s − i·34-s + i·36-s + (1 + i)37-s + (1 + i)41-s − i·49-s + (1 − i)58-s + ⋯
L(s)  = 1  − 2-s + 4-s − 8-s + i·9-s + 16-s + i·17-s − i·18-s + (−1 + i)29-s − 32-s − i·34-s + i·36-s + (1 + i)37-s + (1 + i)41-s − i·49-s + (1 − i)58-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.429 - 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.429 - 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1700\)    =    \(2^{2} \cdot 5^{2} \cdot 17\)
Sign: $0.429 - 0.902i$
Analytic conductor: \(0.848410\)
Root analytic conductor: \(0.921092\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1700} (599, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1700,\ (\ :0),\ 0.429 - 0.902i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6662640302\)
\(L(\frac12)\) \(\approx\) \(0.6662640302\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
5 \( 1 \)
17 \( 1 - iT \)
good3 \( 1 - iT^{2} \)
7 \( 1 + iT^{2} \)
11 \( 1 - iT^{2} \)
13 \( 1 - T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 + (1 - i)T - iT^{2} \)
31 \( 1 + iT^{2} \)
37 \( 1 + (-1 - i)T + iT^{2} \)
41 \( 1 + (-1 - i)T + iT^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 + (1 + i)T + iT^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + iT^{2} \)
73 \( 1 + (-1 - i)T + iT^{2} \)
79 \( 1 - iT^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (-1 - i)T + iT^{2} \)
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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

−9.675973171610741880002485233631, −8.839197949957727038205227031445, −8.079531703049957269561222649299, −7.57743247305971626789641472862, −6.61146551232372573813698466420, −5.82836971516721057631842478349, −4.84714458698685385258314122428, −3.57656010934202825583265471919, −2.45180026866255025283259880845, −1.47283931014291734955803349509, 0.73529168076450384789291264501, 2.19685911830970152693719719983, 3.20387380085439095013493761912, 4.26212232724169480419943684653, 5.67228740783405640674547744168, 6.26159094303157856638113848005, 7.27671312364989155538289425095, 7.70770368266227088634182141654, 8.864938947533954012293387069381, 9.316078711510165378088038248835

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