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Properties

Label 2-1700-85.4-c1-0-17
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: $\chi_{1700} (1449, \cdot )$
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

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

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