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Properties

Label 2-3400-680.259-c0-0-2
Degree $2$
Conductor $3400$
Sign $0.902 + 0.429i$
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  + 2-s + (−1 − i)3-s + 4-s + (−1 − i)6-s + 8-s + i·9-s + (1 + i)11-s + (−1 − i)12-s + 16-s + 17-s + i·18-s + 2i·19-s + (1 + i)22-s + (−1 − i)24-s + 32-s − 2i·33-s + ⋯
L(s)  = 1  + 2-s + (−1 − i)3-s + 4-s + (−1 − i)6-s + 8-s + i·9-s + (1 + i)11-s + (−1 − i)12-s + 16-s + 17-s + i·18-s + 2i·19-s + (1 + i)22-s + (−1 − i)24-s + 32-s − 2i·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.902 + 0.429i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.902 + 0.429i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3400\)    =    \(2^{3} \cdot 5^{2} \cdot 17\)
Sign: $0.902 + 0.429i$
Analytic conductor: \(1.69682\)
Root analytic conductor: \(1.30262\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3400} (2299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3400,\ (\ :0),\ 0.902 + 0.429i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.945272207\)
\(L(\frac12)\) \(\approx\) \(1.945272207\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−8.431224682150013529108607587560, −7.53978018297950376832477941522, −7.09976776313948135979500678531, −6.34804753083369111674632538790, −5.78855039686796265003335310973, −5.14780402585633515496536468858, −4.11341654229099925768809742684, −3.39768213304967656533991773737, −1.90726839944321565253541722163, −1.39218812189018332306246594231, 1.12651626377619686095662212557, 2.76580015199538129272136906220, 3.56448364837946694602672571706, 4.35438413167123921694512578717, 5.01108322667315234370289882424, 5.65778406117974507217775836602, 6.34565354833593687130139091752, 6.95814448442209925446577500405, 7.999414654764499291161302299081, 9.003011315011733286897202263854

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