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Properties

Label 2-1700-340.319-c0-0-1
Degree $2$
Conductor $1700$
Sign $0.980 + 0.197i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 + 0.197i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 + 0.197i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1700\)    =    \(2^{2} \cdot 5^{2} \cdot 17\)
Sign: $0.980 + 0.197i$
Analytic conductor: \(0.848410\)
Root analytic conductor: \(0.921092\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1700} (999, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1700,\ (\ :0),\ 0.980 + 0.197i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.138919489\)
\(L(\frac12)\) \(\approx\) \(2.138919489\)
\(L(1)\) 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

−9.596393949385598160218657092051, −8.669445881447363507521402166728, −7.73192702609329933456227232365, −6.92642315866123397806888009250, −6.11545153914844352516051261662, −5.56736843597436082798746932895, −4.34440511225591946248593006061, −3.75552775420065719132237302327, −2.76184253561229393790250649247, −1.51286033130459210392655181860, 1.73582178015030478815589437296, 2.72249878304422323304385500755, 3.68906659317321237553215472093, 4.76777340488987016072813203585, 5.28316851115856781295363667617, 6.18659364125853403226402685609, 7.23091686453802682774133472424, 7.61672925377273619823878620305, 8.693717107571093740018188494762, 9.670873880778663769443979775990

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