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Properties

Label 2-3400-680.659-c0-0-4
Degree $2$
Conductor $3400$
Sign $-0.197 + 0.980i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 3400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.197 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.197 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3400\)    =    \(2^{3} \cdot 5^{2} \cdot 17\)
Sign: $-0.197 + 0.980i$
Analytic conductor: \(1.69682\)
Root analytic conductor: \(1.30262\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3400} (2699, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3400,\ (\ :0),\ -0.197 + 0.980i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.149947754\)
\(L(\frac12)\) \(\approx\) \(1.149947754\)
\(L(1)\) 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

−8.652315077141616241069520373513, −8.010109674442084828369420628430, −7.12866398866948493251909583956, −6.71311567273992455936203853584, −6.06127419978028819299755566357, −4.70524820031684273215584093117, −3.38807882715577046615854375165, −2.73230140741530695782723455914, −1.86775558845767054019636798621, −0.844557040083936034175625390919, 1.63803324554598196518144561821, 2.38056493411393368941019534902, 3.60220289743565585917969749479, 3.99205179042792409060250874781, 5.12507247180082547333880838115, 6.26306008645796420053645587707, 6.92833690519049600528407255557, 7.80780056785084087366071227441, 8.473820128804237508209889326189, 9.032883690035726614863344845406

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