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Properties

Label 2-40e2-20.3-c1-0-23
Degree $2$
Conductor $1600$
Sign $0.793 + 0.608i$
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  + (2.22 − 2.22i)3-s + (2 + 2i)7-s − 6.89i·9-s + 3.44i·11-s + (4.44 + 4.44i)13-s + (0.775 − 0.775i)17-s + 2.55·19-s + 8.89·21-s + (0.449 − 0.449i)23-s + (−8.67 − 8.67i)27-s + 2.89i·29-s − 2.89i·31-s + (7.67 + 7.67i)33-s + (−6 + 6i)37-s + 19.7·39-s + ⋯
L(s)  = 1  + (1.28 − 1.28i)3-s + (0.755 + 0.755i)7-s − 2.29i·9-s + 1.04i·11-s + (1.23 + 1.23i)13-s + (0.188 − 0.188i)17-s + 0.585·19-s + 1.94·21-s + (0.0937 − 0.0937i)23-s + (−1.66 − 1.66i)27-s + 0.538i·29-s − 0.520i·31-s + (1.33 + 1.33i)33-s + (−0.986 + 0.986i)37-s + 3.17·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.793 + 0.608i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.793 + 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $0.793 + 0.608i$
Analytic conductor: \(12.7760\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (1343, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ 0.793 + 0.608i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.100310154\)
\(L(\frac12)\) \(\approx\) \(3.100310154\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.22 + 2.22i)T - 3iT^{2} \)
7 \( 1 + (-2 - 2i)T + 7iT^{2} \)
11 \( 1 - 3.44iT - 11T^{2} \)
13 \( 1 + (-4.44 - 4.44i)T + 13iT^{2} \)
17 \( 1 + (-0.775 + 0.775i)T - 17iT^{2} \)
19 \( 1 - 2.55T + 19T^{2} \)
23 \( 1 + (-0.449 + 0.449i)T - 23iT^{2} \)
29 \( 1 - 2.89iT - 29T^{2} \)
31 \( 1 + 2.89iT - 31T^{2} \)
37 \( 1 + (6 - 6i)T - 37iT^{2} \)
41 \( 1 - 5T + 41T^{2} \)
43 \( 1 + (-2.89 + 2.89i)T - 43iT^{2} \)
47 \( 1 + (7.34 + 7.34i)T + 47iT^{2} \)
53 \( 1 + (6.44 + 6.44i)T + 53iT^{2} \)
59 \( 1 + 7.79T + 59T^{2} \)
61 \( 1 - 0.898T + 61T^{2} \)
67 \( 1 + (-4.22 - 4.22i)T + 67iT^{2} \)
71 \( 1 + 2iT - 71T^{2} \)
73 \( 1 + (5.67 + 5.67i)T + 73iT^{2} \)
79 \( 1 + 0.898T + 79T^{2} \)
83 \( 1 + (-4.67 + 4.67i)T - 83iT^{2} \)
89 \( 1 - 11.8iT - 89T^{2} \)
97 \( 1 + (4.89 - 4.89i)T - 97iT^{2} \)
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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

−9.091039366170406802979717197388, −8.453505709398402588530818637872, −7.86168330803686036652909530590, −6.98221568294987898347919165209, −6.46125310710533245818852460470, −5.24793839272349673776392682753, −4.07474417331806043410335434127, −3.04933614909202244520993750939, −1.95543136562099562637985653098, −1.48390110493691424403311188897, 1.27457257402089551913919111158, 2.88759356800903112260310520184, 3.49644767181320993648663528174, 4.23519811919256876401453829141, 5.18191472582528517818001868250, 6.03476688401065034078046344679, 7.62403552638214337504783316821, 8.023678455791944725684128104908, 8.692263709844719815837247998274, 9.394215593588823557824021317482

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