Properties

Label 2-3328-8.5-c1-0-66
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  − 2.56i·3-s + 0.561i·5-s + 0.561·7-s − 3.56·9-s + 2i·11-s + i·13-s + 1.43·15-s − 0.561·17-s − 6i·19-s − 1.43i·21-s + 4.68·25-s + 1.43i·27-s + 8.24i·29-s + 7.12·31-s + 5.12·33-s + ⋯
L(s)  = 1  − 1.47i·3-s + 0.251i·5-s + 0.212·7-s − 1.18·9-s + 0.603i·11-s + 0.277i·13-s + 0.371·15-s − 0.136·17-s − 1.37i·19-s − 0.313i·21-s + 0.936·25-s + 0.276i·27-s + 1.53i·29-s + 1.27·31-s + 0.891·33-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

\(L(1)\) \(\approx\) \(1.583160922\)
\(L(\frac12)\) \(\approx\) \(1.583160922\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.56iT - 3T^{2} \)
5 \( 1 - 0.561iT - 5T^{2} \)
7 \( 1 - 0.561T + 7T^{2} \)
11 \( 1 - 2iT - 11T^{2} \)
17 \( 1 + 0.561T + 17T^{2} \)
19 \( 1 + 6iT - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 8.24iT - 29T^{2} \)
31 \( 1 - 7.12T + 31T^{2} \)
37 \( 1 + 9.68iT - 37T^{2} \)
41 \( 1 + 7.12T + 41T^{2} \)
43 \( 1 + 8.80iT - 43T^{2} \)
47 \( 1 - 1.68T + 47T^{2} \)
53 \( 1 + 4.87iT - 53T^{2} \)
59 \( 1 + 6iT - 59T^{2} \)
61 \( 1 + 13.3iT - 61T^{2} \)
67 \( 1 + 6iT - 67T^{2} \)
71 \( 1 - 1.68T + 71T^{2} \)
73 \( 1 + 10T + 73T^{2} \)
79 \( 1 - 12T + 79T^{2} \)
83 \( 1 - 17.3iT - 83T^{2} \)
89 \( 1 - 8.24T + 89T^{2} \)
97 \( 1 + 6T + 97T^{2} \)
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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

−8.258267956309440298189344773387, −7.41380988213951001056374437032, −6.78691761941958015226971518371, −6.58436370184192671768946788196, −5.31906197601155050882861217555, −4.67018708885454969078376339697, −3.37233861712294229574377244920, −2.40497048353405973885080917048, −1.68633819706940691267643657240, −0.51953041496009269758982434890, 1.20591037136246440459546610644, 2.76469441678361015706436958863, 3.48859864704244867836136604446, 4.41018675236551251944725689463, 4.84482740673248869460478573687, 5.80921604535327286293799612255, 6.37436671673358214017311376735, 7.68303075439633656023457329622, 8.380778142042603720107616909486, 8.862545597647608980596184119813

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