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Properties

Label 2-3328-8.5-c1-0-53
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  − 2i·5-s − 2·7-s + 3·9-s − 2i·11-s − i·13-s + 6·17-s + 6i·19-s + 8·23-s + 25-s + 2i·29-s − 10·31-s + 4i·35-s + 6i·37-s + 6·41-s + 4i·43-s + ⋯
L(s)  = 1  − 0.894i·5-s − 0.755·7-s + 9-s − 0.603i·11-s − 0.277i·13-s + 1.45·17-s + 1.37i·19-s + 1.66·23-s + 0.200·25-s + 0.371i·29-s − 1.79·31-s + 0.676i·35-s + 0.986i·37-s + 0.937·41-s + 0.609i·43-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

\(L(1)\) \(\approx\) \(1.981535823\)
\(L(\frac12)\) \(\approx\) \(1.981535823\)
\(L(\frac{3}{2})\) 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 \( 1 - 3T^{2} \)
5 \( 1 + 2iT - 5T^{2} \)
7 \( 1 + 2T + 7T^{2} \)
11 \( 1 + 2iT - 11T^{2} \)
17 \( 1 - 6T + 17T^{2} \)
19 \( 1 - 6iT - 19T^{2} \)
23 \( 1 - 8T + 23T^{2} \)
29 \( 1 - 2iT - 29T^{2} \)
31 \( 1 + 10T + 31T^{2} \)
37 \( 1 - 6iT - 37T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 - 4iT - 43T^{2} \)
47 \( 1 - 2T + 47T^{2} \)
53 \( 1 + 6iT - 53T^{2} \)
59 \( 1 + 10iT - 59T^{2} \)
61 \( 1 + 2iT - 61T^{2} \)
67 \( 1 + 10iT - 67T^{2} \)
71 \( 1 - 10T + 71T^{2} \)
73 \( 1 + 2T + 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 - 6iT - 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 - 2T + 97T^{2} \)
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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

−8.501790641516419792698386485765, −7.84246254566311702676034805965, −7.10041375439695171027985830439, −6.24213511517197549276612964489, −5.41428232339832053708538941923, −4.83534112165499395539914685161, −3.64429077591181687790501944794, −3.22506366195855257825503171715, −1.61858460732684329621560685007, −0.810237419234173064450958648354, 0.978048321201144394584188386267, 2.32121538376297524152832952161, 3.14981636149932821211840835164, 3.92811730028493936210430088391, 4.89100168812448932038231351354, 5.74790785468569100893343031948, 6.75804817318721114003935994198, 7.19473966854861786710649131782, 7.57280546991219335579010534867, 9.090322850212685434003947732148

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