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Properties

Label 2-40e2-80.29-c1-0-28
Degree $2$
Conductor $1600$
Sign $-0.798 + 0.601i$
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  + (−0.120 + 0.120i)3-s − 2.66·7-s + 2.97i·9-s + (3.49 − 3.49i)11-s + (−2.94 + 2.94i)13-s − 1.85i·17-s + (−3.44 − 3.44i)19-s + (0.320 − 0.320i)21-s − 0.707·23-s + (−0.716 − 0.716i)27-s + (3.49 + 3.49i)29-s − 6.84·31-s + 0.839i·33-s + (0.0975 + 0.0975i)37-s − 0.705i·39-s + ⋯
L(s)  = 1  + (−0.0692 + 0.0692i)3-s − 1.00·7-s + 0.990i·9-s + (1.05 − 1.05i)11-s + (−0.815 + 0.815i)13-s − 0.448i·17-s + (−0.791 − 0.791i)19-s + (0.0698 − 0.0698i)21-s − 0.147·23-s + (−0.137 − 0.137i)27-s + (0.649 + 0.649i)29-s − 1.22·31-s + 0.146i·33-s + (0.0160 + 0.0160i)37-s − 0.113i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.798 + 0.601i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.798 + 0.601i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $-0.798 + 0.601i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (849, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ -0.798 + 0.601i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3400835372\)
\(L(\frac12)\) \(\approx\) \(0.3400835372\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.120 - 0.120i)T - 3iT^{2} \)
7 \( 1 + 2.66T + 7T^{2} \)
11 \( 1 + (-3.49 + 3.49i)T - 11iT^{2} \)
13 \( 1 + (2.94 - 2.94i)T - 13iT^{2} \)
17 \( 1 + 1.85iT - 17T^{2} \)
19 \( 1 + (3.44 + 3.44i)T + 19iT^{2} \)
23 \( 1 + 0.707T + 23T^{2} \)
29 \( 1 + (-3.49 - 3.49i)T + 29iT^{2} \)
31 \( 1 + 6.84T + 31T^{2} \)
37 \( 1 + (-0.0975 - 0.0975i)T + 37iT^{2} \)
41 \( 1 + 10.2iT - 41T^{2} \)
43 \( 1 + (4.43 + 4.43i)T + 43iT^{2} \)
47 \( 1 + 1.89iT - 47T^{2} \)
53 \( 1 + (7.43 + 7.43i)T + 53iT^{2} \)
59 \( 1 + (-0.959 + 0.959i)T - 59iT^{2} \)
61 \( 1 + (-6.49 - 6.49i)T + 61iT^{2} \)
67 \( 1 + (3.49 - 3.49i)T - 67iT^{2} \)
71 \( 1 + 7.86iT - 71T^{2} \)
73 \( 1 + 15.6T + 73T^{2} \)
79 \( 1 + 6.70T + 79T^{2} \)
83 \( 1 + (3.87 - 3.87i)T - 83iT^{2} \)
89 \( 1 + 10.5iT - 89T^{2} \)
97 \( 1 + 4.79iT - 97T^{2} \)
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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

−9.002580857164853505694739010175, −8.572872415533904115477326346001, −7.23153239789946986832019148659, −6.79936714364688161556586899966, −5.86779303943658249306932141284, −4.93764736334957423934353616276, −3.98508206315005750528380935573, −3.01469574996830061441397194952, −1.91251806119498569155396533461, −0.12859219691612316247449684928, 1.50052804003840405193782270783, 2.88043535240100912161034677284, 3.79456077203425785362106747860, 4.59937012763693306053847440686, 5.94801629112516033413509982878, 6.43557494645401032746327432061, 7.19108872528605524673397590827, 8.122280988899160931946522365913, 9.120826880275057229364037846666, 9.819223046109688932895073095203

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