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Properties

Label 2-3400-136.19-c0-0-3
Degree $2$
Conductor $3400$
Sign $-0.0465 + 0.998i$
Analytic cond. $1.69682$
Root an. cond. $1.30262$
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  + (0.707 + 0.707i)2-s + (−0.707 + 0.292i)3-s + 1.00i·4-s + (−0.707 − 0.292i)6-s + (−0.707 + 0.707i)8-s + (−0.292 + 0.292i)9-s + (−1.70 − 0.707i)11-s + (−0.292 − 0.707i)12-s − 1.00·16-s + (−0.707 − 0.707i)17-s − 0.414·18-s + (−0.707 − 1.70i)22-s + (0.292 − 0.707i)24-s + (0.414 − 1.00i)27-s + (−0.707 − 0.707i)32-s + 1.41·33-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)2-s + (−0.707 + 0.292i)3-s + 1.00i·4-s + (−0.707 − 0.292i)6-s + (−0.707 + 0.707i)8-s + (−0.292 + 0.292i)9-s + (−1.70 − 0.707i)11-s + (−0.292 − 0.707i)12-s − 1.00·16-s + (−0.707 − 0.707i)17-s − 0.414·18-s + (−0.707 − 1.70i)22-s + (0.292 − 0.707i)24-s + (0.414 − 1.00i)27-s + (−0.707 − 0.707i)32-s + 1.41·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0465 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0465 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3400\)    =    \(2^{3} \cdot 5^{2} \cdot 17\)
Sign: $-0.0465 + 0.998i$
Analytic conductor: \(1.69682\)
Root analytic conductor: \(1.30262\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3400} (1651, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3400,\ (\ :0),\ -0.0465 + 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.008085292870\)
\(L(\frac12)\) \(\approx\) \(0.008085292870\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 0.707i)T \)
5 \( 1 \)
17 \( 1 + (0.707 + 0.707i)T \)
good3 \( 1 + (0.707 - 0.292i)T + (0.707 - 0.707i)T^{2} \)
7 \( 1 + (0.707 + 0.707i)T^{2} \)
11 \( 1 + (1.70 + 0.707i)T + (0.707 + 0.707i)T^{2} \)
13 \( 1 + T^{2} \)
19 \( 1 + iT^{2} \)
23 \( 1 + (-0.707 - 0.707i)T^{2} \)
29 \( 1 + (0.707 - 0.707i)T^{2} \)
31 \( 1 + (-0.707 + 0.707i)T^{2} \)
37 \( 1 + (-0.707 + 0.707i)T^{2} \)
41 \( 1 + (-0.707 + 1.70i)T + (-0.707 - 0.707i)T^{2} \)
43 \( 1 + (1 - i)T - iT^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 + (1 - i)T - iT^{2} \)
61 \( 1 + (0.707 + 0.707i)T^{2} \)
67 \( 1 + 1.41T + T^{2} \)
71 \( 1 + (-0.707 + 0.707i)T^{2} \)
73 \( 1 + (-0.707 - 1.70i)T + (-0.707 + 0.707i)T^{2} \)
79 \( 1 + (-0.707 - 0.707i)T^{2} \)
83 \( 1 + (1 + i)T + iT^{2} \)
89 \( 1 - 1.41iT - T^{2} \)
97 \( 1 + (-0.292 - 0.707i)T + (-0.707 + 0.707i)T^{2} \)
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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

−8.351590547691534425763346151875, −7.80011103946324572353043577348, −7.00557662080613271662733740358, −6.12999912127431254167736732028, −5.46066144457225593229576340454, −5.03512986608381683533828443412, −4.24090313933603841967907592010, −3.05481897243316591248805995528, −2.42286670292168443966877268552, −0.00378463648464126786145330322, 1.58319744937180768030763741154, 2.57181037402506321190495519379, 3.38292428881621134958502910330, 4.58667568838844842706891805006, 5.03517576785247963810457908080, 5.92597096802915587190566166806, 6.42066105256420610390365945590, 7.33899643189524274573950658753, 8.212089361835041869066200653653, 9.153569465293884717773326699823

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