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Properties

Label 2-40e2-16.13-c1-0-27
Degree $2$
Conductor $1600$
Sign $-0.986 - 0.163i$
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  + (−1.03 − 1.03i)3-s + 1.49i·7-s − 0.836i·9-s + (−0.423 + 0.423i)11-s + (1.85 + 1.85i)13-s − 6.50·17-s + (1.75 + 1.75i)19-s + (1.55 − 1.55i)21-s − 7.19i·23-s + (−3.99 + 3.99i)27-s + (−6.57 − 6.57i)29-s + 6.75·31-s + 0.880·33-s + (−1.95 + 1.95i)37-s − 3.86i·39-s + ⋯
L(s)  = 1  + (−0.600 − 0.600i)3-s + 0.565i·7-s − 0.278i·9-s + (−0.127 + 0.127i)11-s + (0.515 + 0.515i)13-s − 1.57·17-s + (0.403 + 0.403i)19-s + (0.339 − 0.339i)21-s − 1.49i·23-s + (−0.767 + 0.767i)27-s + (−1.22 − 1.22i)29-s + 1.21·31-s + 0.153·33-s + (−0.321 + 0.321i)37-s − 0.618i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.986 - 0.163i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.986 - 0.163i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $-0.986 - 0.163i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (401, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ -0.986 - 0.163i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1948151469\)
\(L(\frac12)\) \(\approx\) \(0.1948151469\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.03 + 1.03i)T + 3iT^{2} \)
7 \( 1 - 1.49iT - 7T^{2} \)
11 \( 1 + (0.423 - 0.423i)T - 11iT^{2} \)
13 \( 1 + (-1.85 - 1.85i)T + 13iT^{2} \)
17 \( 1 + 6.50T + 17T^{2} \)
19 \( 1 + (-1.75 - 1.75i)T + 19iT^{2} \)
23 \( 1 + 7.19iT - 23T^{2} \)
29 \( 1 + (6.57 + 6.57i)T + 29iT^{2} \)
31 \( 1 - 6.75T + 31T^{2} \)
37 \( 1 + (1.95 - 1.95i)T - 37iT^{2} \)
41 \( 1 + 7.70iT - 41T^{2} \)
43 \( 1 + (6.13 - 6.13i)T - 43iT^{2} \)
47 \( 1 + 6.65T + 47T^{2} \)
53 \( 1 + (5.29 - 5.29i)T - 53iT^{2} \)
59 \( 1 + (5.91 - 5.91i)T - 59iT^{2} \)
61 \( 1 + (1.43 + 1.43i)T + 61iT^{2} \)
67 \( 1 + (6.35 + 6.35i)T + 67iT^{2} \)
71 \( 1 + 4.08iT - 71T^{2} \)
73 \( 1 + 2.43iT - 73T^{2} \)
79 \( 1 + 11.6T + 79T^{2} \)
83 \( 1 + (-2.81 - 2.81i)T + 83iT^{2} \)
89 \( 1 - 10.5iT - 89T^{2} \)
97 \( 1 + 18.1T + 97T^{2} \)
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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

−8.952623225771244117026222573980, −8.249351588105103245792561752248, −7.22408696985592552393782189198, −6.36081962713727430527849448843, −6.07030026590745902638972531465, −4.86402855548980588860959928846, −4.00188165624264893422714052306, −2.66271048269007699597306245368, −1.60824819146909461756682557328, −0.079876472159615510897191189477, 1.61577287097955448902983249498, 3.08575973808053738301926959332, 4.05979921832741323635659971601, 4.92730753177875301735039282529, 5.57930095197947308644070098621, 6.58173809699038663676488575838, 7.39129991806317193576158433337, 8.255584140545790360647236436414, 9.131668867977973596904087529767, 9.954566959502892791097611598777

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