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Properties

Label 2-1700-85.64-c1-0-9
Degree $2$
Conductor $1700$
Sign $-0.813 - 0.581i$
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  + (−2.30 + 2.30i)3-s + (2.30 + 2.30i)7-s − 7.60i·9-s + (−0.302 + 0.302i)11-s − 2.60i·13-s + (2 + 3.60i)17-s + 0.605i·19-s − 10.6·21-s + (4.30 + 4.30i)23-s + (10.6 + 10.6i)27-s + (1.60 + 1.60i)29-s + (4.30 + 4.30i)31-s − 1.39i·33-s + (−3 + 3i)37-s + (6 + 6i)39-s + ⋯
L(s)  = 1  + (−1.32 + 1.32i)3-s + (0.870 + 0.870i)7-s − 2.53i·9-s + (−0.0912 + 0.0912i)11-s − 0.722i·13-s + (0.485 + 0.874i)17-s + 0.138i·19-s − 2.31·21-s + (0.897 + 0.897i)23-s + (2.04 + 2.04i)27-s + (0.298 + 0.298i)29-s + (0.772 + 0.772i)31-s − 0.242i·33-s + (−0.493 + 0.493i)37-s + (0.960 + 0.960i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.813 - 0.581i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.813 - 0.581i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1700\)    =    \(2^{2} \cdot 5^{2} \cdot 17\)
Sign: $-0.813 - 0.581i$
Analytic conductor: \(13.5745\)
Root analytic conductor: \(3.68436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1700} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1700,\ (\ :1/2),\ -0.813 - 0.581i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.045500136\)
\(L(\frac12)\) \(\approx\) \(1.045500136\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2 - 3.60i)T \)
good3 \( 1 + (2.30 - 2.30i)T - 3iT^{2} \)
7 \( 1 + (-2.30 - 2.30i)T + 7iT^{2} \)
11 \( 1 + (0.302 - 0.302i)T - 11iT^{2} \)
13 \( 1 + 2.60iT - 13T^{2} \)
19 \( 1 - 0.605iT - 19T^{2} \)
23 \( 1 + (-4.30 - 4.30i)T + 23iT^{2} \)
29 \( 1 + (-1.60 - 1.60i)T + 29iT^{2} \)
31 \( 1 + (-4.30 - 4.30i)T + 31iT^{2} \)
37 \( 1 + (3 - 3i)T - 37iT^{2} \)
41 \( 1 + (-1 + i)T - 41iT^{2} \)
43 \( 1 - 3.39T + 43T^{2} \)
47 \( 1 + 4iT - 47T^{2} \)
53 \( 1 - 5.21T + 53T^{2} \)
59 \( 1 + 8.60iT - 59T^{2} \)
61 \( 1 + (6.21 - 6.21i)T - 61iT^{2} \)
67 \( 1 - 9.21iT - 67T^{2} \)
71 \( 1 + (-2.90 - 2.90i)T + 71iT^{2} \)
73 \( 1 + (7 - 7i)T - 73iT^{2} \)
79 \( 1 + (-0.302 + 0.302i)T - 79iT^{2} \)
83 \( 1 + 17.8T + 83T^{2} \)
89 \( 1 + 7.81T + 89T^{2} \)
97 \( 1 + (-7.60 + 7.60i)T - 97iT^{2} \)
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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

−9.944498258395982737626133235758, −8.919531898706831704382270710349, −8.347640908185483928482189315058, −7.12073884898698697283517499239, −6.04685341136771502362123895661, −5.42143172296256405286973380784, −4.98118680634308648314880820159, −4.00737679954186470530097284791, −3.01012924804501929886288406761, −1.24928660607741693780160164755, 0.56628217038670712899444473794, 1.43408404678366535252819980611, 2.59223156567416492676261322804, 4.39886652455143347172550514407, 4.94910573824371031278311635121, 5.91798297935031790610582591379, 6.68174733899775756522506514174, 7.38321587795640006537612563873, 7.81423155625805755507995289717, 8.850058098915893969900076327910

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