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Properties

Label 2-3400-680.19-c0-0-3
Degree $2$
Conductor $3400$
Sign $0.914 - 0.405i$
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 + (0.292 + 0.707i)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 + (0.707 − 0.292i)12-s − 1.00·16-s + (0.707 − 0.707i)17-s + 0.414i·18-s + (1.70 − 0.707i)22-s + (−0.292 + 0.707i)24-s + (1.00 + 0.414i)27-s + (0.707 − 0.707i)32-s − 1.41i·33-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s + (0.292 + 0.707i)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 + (0.707 − 0.292i)12-s − 1.00·16-s + (0.707 − 0.707i)17-s + 0.414i·18-s + (1.70 − 0.707i)22-s + (−0.292 + 0.707i)24-s + (1.00 + 0.414i)27-s + (0.707 − 0.707i)32-s − 1.41i·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.914 - 0.405i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.914 - 0.405i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3400\)    =    \(2^{3} \cdot 5^{2} \cdot 17\)
Sign: $0.914 - 0.405i$
Analytic conductor: \(1.69682\)
Root analytic conductor: \(1.30262\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3400} (699, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3400,\ (\ :0),\ 0.914 - 0.405i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8957982482\)
\(L(\frac12)\) \(\approx\) \(0.8957982482\)
\(L(1)\) 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 \( 1 + (-0.292 - 0.707i)T + (-0.707 + 0.707i)T^{2} \)
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 \( 1 + 1.41iT - T^{2} \)
71 \( 1 + (-0.707 + 0.707i)T^{2} \)
73 \( 1 + (-1.70 + 0.707i)T + (0.707 - 0.707i)T^{2} \)
79 \( 1 + (-0.707 - 0.707i)T^{2} \)
83 \( 1 + (1 - i)T - iT^{2} \)
89 \( 1 + 1.41iT - T^{2} \)
97 \( 1 + (0.707 - 0.292i)T + (0.707 - 0.707i)T^{2} \)
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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

−8.895223717894265620134420790791, −8.023654946720168537836947642042, −7.58983918596884372308309658829, −6.73363288189377174517824263205, −5.73036887666464993780529518741, −5.23899998126170833723437406820, −4.38654602114121068591389293536, −3.27415534161619276789782531712, −2.35467586670270262547899056378, −0.75547485993573774858481642200, 1.14687571316746639563512534176, 2.23577996918959042715700732826, 2.70258654741254937678507315889, 3.89060381978300455726964237161, 4.79594768360048308732353882344, 5.73377678534188704848748180759, 6.96667749625944477332008706224, 7.45038468891526966829996545187, 8.071344655122407286328542688262, 8.514259213443642530443287510975

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