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Properties

Label 2-1700-17.13-c1-0-13
Degree $2$
Conductor $1700$
Sign $0.390 + 0.920i$
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  + (−1.30 − 1.30i)3-s + (1.30 − 1.30i)7-s + 0.394i·9-s + (3.30 − 3.30i)11-s + 4.60·13-s + (3.60 + 2i)17-s + 6.60i·19-s − 3.39·21-s + (0.697 − 0.697i)23-s + (−3.39 + 3.39i)27-s + (5.60 + 5.60i)29-s + (0.697 + 0.697i)31-s − 8.60·33-s + (−3 − 3i)37-s + (−6 − 6i)39-s + ⋯
L(s)  = 1  + (−0.752 − 0.752i)3-s + (0.492 − 0.492i)7-s + 0.131i·9-s + (0.995 − 0.995i)11-s + 1.27·13-s + (0.874 + 0.485i)17-s + 1.51i·19-s − 0.740·21-s + (0.145 − 0.145i)23-s + (−0.653 + 0.653i)27-s + (1.04 + 1.04i)29-s + (0.125 + 0.125i)31-s − 1.49·33-s + (−0.493 − 0.493i)37-s + (−0.960 − 0.960i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.390 + 0.920i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.390 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1700\)    =    \(2^{2} \cdot 5^{2} \cdot 17\)
Sign: $0.390 + 0.920i$
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: \((2,\ 1700,\ (\ :1/2),\ 0.390 + 0.920i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.659694507\)
\(L(\frac12)\) \(\approx\) \(1.659694507\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.60 - 2i)T \)
good3 \( 1 + (1.30 + 1.30i)T + 3iT^{2} \)
7 \( 1 + (-1.30 + 1.30i)T - 7iT^{2} \)
11 \( 1 + (-3.30 + 3.30i)T - 11iT^{2} \)
13 \( 1 - 4.60T + 13T^{2} \)
19 \( 1 - 6.60iT - 19T^{2} \)
23 \( 1 + (-0.697 + 0.697i)T - 23iT^{2} \)
29 \( 1 + (-5.60 - 5.60i)T + 29iT^{2} \)
31 \( 1 + (-0.697 - 0.697i)T + 31iT^{2} \)
37 \( 1 + (3 + 3i)T + 37iT^{2} \)
41 \( 1 + (-1 + i)T - 41iT^{2} \)
43 \( 1 + 10.6iT - 43T^{2} \)
47 \( 1 - 4T + 47T^{2} \)
53 \( 1 - 9.21iT - 53T^{2} \)
59 \( 1 - 1.39iT - 59T^{2} \)
61 \( 1 + (-8.21 + 8.21i)T - 61iT^{2} \)
67 \( 1 - 5.21T + 67T^{2} \)
71 \( 1 + (7.90 + 7.90i)T + 71iT^{2} \)
73 \( 1 + (-7 - 7i)T + 73iT^{2} \)
79 \( 1 + (-3.30 + 3.30i)T - 79iT^{2} \)
83 \( 1 + 3.81iT - 83T^{2} \)
89 \( 1 + 13.8T + 89T^{2} \)
97 \( 1 + (-0.394 - 0.394i)T + 97iT^{2} \)
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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

−8.953792004987356078053316576682, −8.403191212750497604013312174540, −7.53144765425473904838188841470, −6.64230538694802688318156617282, −6.02237651142808021601246794975, −5.44214036530475382170120297811, −3.97089936921778658944041973640, −3.44588086292963302261888154958, −1.50674370300755918890993303413, −0.972786658972920519533956708899, 1.12217667386095782879300318619, 2.49907798547975550027079614351, 3.83488582149423904868943609473, 4.65000244337550002582243752667, 5.24876597888192165262978354851, 6.19572078961390732759374260850, 6.92278125804756830008080907406, 8.014384107565596967850484925383, 8.792724614166375605365065482034, 9.642439298652235447914068290437

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