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Properties

Label 2-40e2-40.27-c1-0-14
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: $\chi_{1600} (607, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1/2),\ 0.973 - 0.229i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.134961555\)
\(L(\frac12)\) \(\approx\) \(2.134961555\)
\(L(\frac{3}{2})\) 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

−9.335471871715548963134810547260, −8.379833289083604216675323831739, −7.84634544188999616789983521419, −7.37277993114567478708472277995, −6.18081979363828298268674060771, −5.42686353838499235316846106494, −4.32105652789930491658332492539, −3.37673221799184133562580383159, −2.22009061319121592116103461872, −1.32389948059462705641449834747, 0.868681328818889429323508678969, 2.76498067237927063748406916396, 3.04649248551751460664454761990, 4.42221592988129295533729913592, 5.14303012029710053545229930093, 6.00074081911287784179597347611, 7.11121889197624676284593511399, 7.957129164992029050136585001242, 8.710792514662502558128758756072, 9.199986036096361040644173445779

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