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Properties

Label 2-40e2-40.3-c1-0-13
Degree $2$
Conductor $1600$
Sign $0.973 + 0.229i$
Analytic cond. $12.7760$
Root an. cond. $3.57436$
Motivic weight $1$
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 + 4·11-s + (−3 + 3i)13-s + (3 − 3i)17-s + 6i·19-s − 2i·21-s + (3 − 3i)23-s + (−4 + 4i)27-s − 2·29-s + 6i·31-s + (−4 − 4i)33-s + (−3 − 3i)37-s + 6·39-s + ⋯
L(s)  = 1  + (−0.577 − 0.577i)3-s + (0.377 + 0.377i)7-s − 0.333i·9-s + 1.20·11-s + (−0.832 + 0.832i)13-s + (0.727 − 0.727i)17-s + 1.37i·19-s − 0.436i·21-s + (0.625 − 0.625i)23-s + (−0.769 + 0.769i)27-s − 0.371·29-s + 1.07i·31-s + (−0.696 − 0.696i)33-s + (−0.493 − 0.493i)37-s + 0.960·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.229i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $0.973 + 0.229i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (543, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ 0.973 + 0.229i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.509645793\)
\(L(\frac12)\) \(\approx\) \(1.509645793\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 + i)T + 3iT^{2} \)
7 \( 1 + (-1 - i)T + 7iT^{2} \)
11 \( 1 - 4T + 11T^{2} \)
13 \( 1 + (3 - 3i)T - 13iT^{2} \)
17 \( 1 + (-3 + 3i)T - 17iT^{2} \)
19 \( 1 - 6iT - 19T^{2} \)
23 \( 1 + (-3 + 3i)T - 23iT^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 - 6iT - 31T^{2} \)
37 \( 1 + (3 + 3i)T + 37iT^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 + (-3 - 3i)T + 43iT^{2} \)
47 \( 1 + (-9 - 9i)T + 47iT^{2} \)
53 \( 1 + (-5 + 5i)T - 53iT^{2} \)
59 \( 1 + 10iT - 59T^{2} \)
61 \( 1 - 12iT - 61T^{2} \)
67 \( 1 + (-9 + 9i)T - 67iT^{2} \)
71 \( 1 - 6iT - 71T^{2} \)
73 \( 1 + (5 + 5i)T + 73iT^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + (-3 - 3i)T + 83iT^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 + (-7 + 7i)T - 97iT^{2} \)
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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

−9.305393178039716966822555472399, −8.719766673044612478918708286302, −7.53445006873851160867671529588, −6.97740468308658975810006489468, −6.17265562401493949478572748374, −5.42856529628178976286610297630, −4.41173621166901332633662406709, −3.41250153430051590262418356790, −2.01597518886371727199565323716, −0.983405959453079691104987812441, 0.866450618518392576208124893054, 2.35349285089622839385830368836, 3.68069322451626743985511515298, 4.48253475421756990229897947276, 5.29255845273448946373582166915, 5.98821861704053599551446532266, 7.17467628510064910004177427988, 7.67933377654365724977655464813, 8.779328227780187777344334953433, 9.533014053654160072655803997924

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