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Properties

Label 2-1700-17.13-c1-0-21
Degree $2$
Conductor $1700$
Sign $0.884 + 0.465i$
Analytic cond. $13.5745$
Root an. cond. $3.68436$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.11 + 1.11i)3-s + (0.260 − 0.260i)7-s − 0.495i·9-s + (3.25 − 3.25i)11-s − 1.91·13-s + (4.05 − 0.731i)17-s − 7.60i·19-s + 0.581·21-s + (−4.56 + 4.56i)23-s + (3.91 − 3.91i)27-s + (1.03 + 1.03i)29-s + (−3.37 − 3.37i)31-s + 7.28·33-s + (−4.15 − 4.15i)37-s + (−2.14 − 2.14i)39-s + ⋯
L(s)  = 1  + (0.646 + 0.646i)3-s + (0.0982 − 0.0982i)7-s − 0.165i·9-s + (0.981 − 0.981i)11-s − 0.532·13-s + (0.984 − 0.177i)17-s − 1.74i·19-s + 0.126·21-s + (−0.951 + 0.951i)23-s + (0.752 − 0.752i)27-s + (0.191 + 0.191i)29-s + (−0.606 − 0.606i)31-s + 1.26·33-s + (−0.683 − 0.683i)37-s + (−0.343 − 0.343i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.884 + 0.465i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.884 + 0.465i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1700\)    =    \(2^{2} \cdot 5^{2} \cdot 17\)
Sign: $0.884 + 0.465i$
Analytic conductor: \(13.5745\)
Root analytic conductor: \(3.68436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1700} (1101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1700,\ (\ :1/2),\ 0.884 + 0.465i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.186967874\)
\(L(\frac12)\) \(\approx\) \(2.186967874\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
17 \( 1 + (-4.05 + 0.731i)T \)
good3 \( 1 + (-1.11 - 1.11i)T + 3iT^{2} \)
7 \( 1 + (-0.260 + 0.260i)T - 7iT^{2} \)
11 \( 1 + (-3.25 + 3.25i)T - 11iT^{2} \)
13 \( 1 + 1.91T + 13T^{2} \)
19 \( 1 + 7.60iT - 19T^{2} \)
23 \( 1 + (4.56 - 4.56i)T - 23iT^{2} \)
29 \( 1 + (-1.03 - 1.03i)T + 29iT^{2} \)
31 \( 1 + (3.37 + 3.37i)T + 31iT^{2} \)
37 \( 1 + (4.15 + 4.15i)T + 37iT^{2} \)
41 \( 1 + (-6.62 + 6.62i)T - 41iT^{2} \)
43 \( 1 - 1.38iT - 43T^{2} \)
47 \( 1 + 0.925T + 47T^{2} \)
53 \( 1 + 9.84iT - 53T^{2} \)
59 \( 1 - 11.2iT - 59T^{2} \)
61 \( 1 + (-0.516 + 0.516i)T - 61iT^{2} \)
67 \( 1 - 12.1T + 67T^{2} \)
71 \( 1 + (-8.06 - 8.06i)T + 71iT^{2} \)
73 \( 1 + (0.904 + 0.904i)T + 73iT^{2} \)
79 \( 1 + (-2.76 + 2.76i)T - 79iT^{2} \)
83 \( 1 + 4.80iT - 83T^{2} \)
89 \( 1 + 3.76T + 89T^{2} \)
97 \( 1 + (-9.10 - 9.10i)T + 97iT^{2} \)
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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

−9.265047203289136991709819644817, −8.737355824300125515435854553129, −7.77174895242495173946783178565, −6.96449368235862233423068674765, −6.00048868877864051945710460109, −5.12156141333760476781239999348, −4.00242116244782329123630978789, −3.47121498828054243719953238259, −2.42525151850291752433060247707, −0.816371125890115339830026280043, 1.46281102434777747700825596046, 2.16348341160995607373496937518, 3.38474597326611712928056972132, 4.32171651461253464147845842957, 5.33672965180863963232333098774, 6.36560165558124583141477099690, 7.10880495657890252141330293577, 7.981360231536234543542963583905, 8.294528876468800727835781829119, 9.463799262544970869650437877260

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