Properties

Label 2-3328-8.5-c1-0-44
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  + 3.16i·3-s + 0.368i·5-s − 2.90·7-s − 7.01·9-s − 1.58i·11-s − i·13-s − 1.16·15-s + 6.69·17-s − 2.80i·19-s − 9.18i·21-s + 5.21·23-s + 4.86·25-s − 12.7i·27-s − 7.21i·29-s − 1.91·31-s + ⋯
L(s)  = 1  + 1.82i·3-s + 0.164i·5-s − 1.09·7-s − 2.33·9-s − 0.477i·11-s − 0.277i·13-s − 0.300·15-s + 1.62·17-s − 0.642i·19-s − 2.00i·21-s + 1.08·23-s + 0.972·25-s − 2.44i·27-s − 1.34i·29-s − 0.343·31-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

\(L(1)\) \(\approx\) \(1.402248402\)
\(L(\frac12)\) \(\approx\) \(1.402248402\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.16iT - 3T^{2} \)
5 \( 1 - 0.368iT - 5T^{2} \)
7 \( 1 + 2.90T + 7T^{2} \)
11 \( 1 + 1.58iT - 11T^{2} \)
17 \( 1 - 6.69T + 17T^{2} \)
19 \( 1 + 2.80iT - 19T^{2} \)
23 \( 1 - 5.21T + 23T^{2} \)
29 \( 1 + 7.21iT - 29T^{2} \)
31 \( 1 + 1.91T + 31T^{2} \)
37 \( 1 + 10.3iT - 37T^{2} \)
41 \( 1 - 2.48T + 41T^{2} \)
43 \( 1 - 6.81iT - 43T^{2} \)
47 \( 1 + 0.219T + 47T^{2} \)
53 \( 1 - 6.48iT - 53T^{2} \)
59 \( 1 + 1.58iT - 59T^{2} \)
61 \( 1 + 10.1iT - 61T^{2} \)
67 \( 1 + 1.32iT - 67T^{2} \)
71 \( 1 + 6.06T + 71T^{2} \)
73 \( 1 + 15.3T + 73T^{2} \)
79 \( 1 - 7.16T + 79T^{2} \)
83 \( 1 - 16.2iT - 83T^{2} \)
89 \( 1 - 10.1T + 89T^{2} \)
97 \( 1 - 4.91T + 97T^{2} \)
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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

−9.055745296242487755583016999688, −8.189833927205918934241330490095, −7.25468401308645756852530897699, −6.16043279460714990303218949117, −5.62292555041603060106871203517, −4.85012145438844563180719629241, −3.96646887313159912159907858735, −3.18687288933447944324524798266, −2.82451208954641793887258234717, −0.55927554402220469330941762884, 0.913881356778431432627693846761, 1.66444780016980621372359193610, 2.91145212789052205191288207586, 3.38010705632519923029679515830, 4.96252527550581905776882726789, 5.76497643772232598500051348570, 6.46002115192695324507905548144, 7.13702848174698914754047313714, 7.48933948480212124288956226178, 8.484162245112469218046943922656

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