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Properties

Label 2-3328-8.5-c1-0-70
Degree $2$
Conductor $3328$
Sign $0.707 + 0.707i$
Analytic cond. $26.5742$
Root an. cond. $5.15501$
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  + 3i·3-s + i·5-s + 7-s − 6·9-s − 2i·11-s − i·13-s − 3·15-s − 3·17-s − 6i·19-s + 3i·21-s − 4·23-s + 4·25-s − 9i·27-s + 2i·29-s − 4·31-s + ⋯
L(s)  = 1  + 1.73i·3-s + 0.447i·5-s + 0.377·7-s − 2·9-s − 0.603i·11-s − 0.277i·13-s − 0.774·15-s − 0.727·17-s − 1.37i·19-s + 0.654i·21-s − 0.834·23-s + 0.800·25-s − 1.73i·27-s + 0.371i·29-s − 0.718·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3328\)    =    \(2^{8} \cdot 13\)
Sign: $0.707 + 0.707i$
Analytic conductor: \(26.5742\)
Root analytic conductor: \(5.15501\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3328} (1665, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3328,\ (\ :1/2),\ 0.707 + 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6378305375\)
\(L(\frac12)\) \(\approx\) \(0.6378305375\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 + iT \)
good3 \( 1 - 3iT - 3T^{2} \)
5 \( 1 - iT - 5T^{2} \)
7 \( 1 - T + 7T^{2} \)
11 \( 1 + 2iT - 11T^{2} \)
17 \( 1 + 3T + 17T^{2} \)
19 \( 1 + 6iT - 19T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 - 2iT - 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + 3iT - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 5iT - 43T^{2} \)
47 \( 1 + 13T + 47T^{2} \)
53 \( 1 + 12iT - 53T^{2} \)
59 \( 1 + 10iT - 59T^{2} \)
61 \( 1 + 8iT - 61T^{2} \)
67 \( 1 - 2iT - 67T^{2} \)
71 \( 1 + 5T + 71T^{2} \)
73 \( 1 - 10T + 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 - 14T + 97T^{2} \)
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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

−8.682994836079994959032275776715, −8.090765901013690424393372391025, −6.92057172453578396373215675618, −6.19239529626136992431261518225, −5.06117248930135294251772947522, −4.89481547447543899874024503085, −3.74943463527758757130336842098, −3.24013323142555566196186392235, −2.20221870453780770187228492094, −0.18792298028208302047658109067, 1.32105363370971559727820856801, 1.83377373724100772236198169320, 2.82150285396636855460024412207, 4.11082707480272096375867131224, 5.01489387838641451493279601178, 5.98416751289220062166517560905, 6.50141692655652913715507710851, 7.32101707708247806601467601059, 7.917871440828012235938256913667, 8.431822158473013193239323477976

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