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Properties

Label 2-3400-136.67-c0-0-10
Degree $2$
Conductor $3400$
Sign $-1$
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  − i·2-s − i·3-s − 4-s − 6-s + i·8-s + i·12-s − i·13-s + 16-s − i·17-s + 19-s + 24-s − 26-s − i·27-s + 29-s − 31-s − i·32-s + ⋯
L(s)  = 1  − i·2-s − i·3-s − 4-s − 6-s + i·8-s + i·12-s − i·13-s + 16-s − i·17-s + 19-s + 24-s − 26-s − i·27-s + 29-s − 31-s − i·32-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3400\)    =    \(2^{3} \cdot 5^{2} \cdot 17\)
Sign: $-1$
Analytic conductor: \(1.69682\)
Root analytic conductor: \(1.30262\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3400} (2651, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3400,\ (\ :0),\ -1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.103682544\)
\(L(\frac12)\) \(\approx\) \(1.103682544\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−8.384575529735926501002882093057, −7.71164586609610340796897231098, −7.15424688169791163877526717660, −6.10166438067682901674595807113, −5.27130744532752529889585701655, −4.53691308171382262788289397432, −3.35175541615927728994161425271, −2.72058404522367616850090343107, −1.65723217219492008190225128345, −0.72433680886773515012673923822, 1.52013431555068982686160366509, 3.21763521507696769708548465033, 4.01831466737695484044654791721, 4.58173420821694290501554814653, 5.36024041351487036104265989203, 6.12228195856269291047744161338, 6.97095403286484449806973081214, 7.56091230370977776668696861534, 8.589805399756294158171615713261, 8.994612408453846349274771766540

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