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Properties

Label 2-40e2-5.2-c0-0-1
Degree $2$
Conductor $1600$
Sign $0.850 + 0.525i$
Analytic cond. $0.798504$
Root an. cond. $0.893590$
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  + (−1 + i)3-s + (−1 − i)7-s − i·9-s + 2·21-s + (1 − i)23-s − 2i·29-s + (1 − i)43-s + (1 + i)47-s + i·49-s + (−1 + i)63-s + (1 + i)67-s + 2i·69-s + 81-s + (1 − i)83-s + (2 + 2i)87-s + ⋯
L(s)  = 1  + (−1 + i)3-s + (−1 − i)7-s − i·9-s + 2·21-s + (1 − i)23-s − 2i·29-s + (1 − i)43-s + (1 + i)47-s + i·49-s + (−1 + i)63-s + (1 + i)67-s + 2i·69-s + 81-s + (1 − i)83-s + (2 + 2i)87-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $0.850 + 0.525i$
Analytic conductor: \(0.798504\)
Root analytic conductor: \(0.893590\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :0),\ 0.850 + 0.525i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5913713371\)
\(L(\frac12)\) \(\approx\) \(0.5913713371\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−9.729460302158698731675019460483, −9.060901116853319297913509992627, −7.85505237748298831364543957271, −6.92340797262602558140441530626, −6.22041960582588081252379114895, −5.43784671243576231369934894367, −4.36706039008445730316710681308, −3.94599839881235936888079873317, −2.70410768916376011757813084711, −0.59147959482019037054521395577, 1.24958978499490677720461180141, 2.56395168842232300890867732186, 3.58261539079594797861871942478, 5.18714801062533176031294069748, 5.60929089008795643751488387041, 6.55986998524535030237384893101, 6.95941736209400434148230776282, 7.917833732876467541241229579751, 9.043928265447382677534423569293, 9.466791997473487130280737475219

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