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Properties

Label 2-3328-13.5-c0-0-3
Degree $2$
Conductor $3328$
Sign $0.289 + 0.957i$
Analytic cond. $1.66088$
Root an. cond. $1.28875$
Motivic weight $0$
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)5-s − 9-s − 13-s − 2i·17-s − i·25-s + 2·29-s + (−1 − i)37-s + (1 − i)41-s + (1 − i)45-s − i·49-s + (1 − i)65-s + (1 + i)73-s + 81-s + (2 + 2i)85-s + (−1 − i)89-s + ⋯
L(s)  = 1  + (−1 + i)5-s − 9-s − 13-s − 2i·17-s − i·25-s + 2·29-s + (−1 − i)37-s + (1 − i)41-s + (1 − i)45-s − i·49-s + (1 − i)65-s + (1 + i)73-s + 81-s + (2 + 2i)85-s + (−1 − i)89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.289 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.289 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3328\)    =    \(2^{8} \cdot 13\)
Sign: $0.289 + 0.957i$
Analytic conductor: \(1.66088\)
Root analytic conductor: \(1.28875\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3328} (3073, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3328,\ (\ :0),\ 0.289 + 0.957i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5753204244\)
\(L(\frac12)\) \(\approx\) \(0.5753204244\)
\(L(1)\) 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 \( 1 + T \)
good3 \( 1 + T^{2} \)
5 \( 1 + (1 - i)T - iT^{2} \)
7 \( 1 + iT^{2} \)
11 \( 1 + iT^{2} \)
17 \( 1 + 2iT - T^{2} \)
19 \( 1 - iT^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - 2T + T^{2} \)
31 \( 1 - iT^{2} \)
37 \( 1 + (1 + i)T + iT^{2} \)
41 \( 1 + (-1 + i)T - iT^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + iT^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 - iT^{2} \)
73 \( 1 + (-1 - i)T + iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + (1 + i)T + iT^{2} \)
97 \( 1 + (1 - i)T - iT^{2} \)
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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

−8.582985594314454595550991724989, −7.80401936377682491840190525021, −7.12030323217604796249396735785, −6.72916288767080884441788063230, −5.53177382797696908474011046011, −4.86653243440099295009878688818, −3.86651587894082261924268524192, −2.88168046444037623653313610715, −2.53719421717224670509687213378, −0.37409227542500695295941811801, 1.18954866088576765547196896422, 2.56133958947634837638968183931, 3.53349567947064005915898373053, 4.43827007065705420078543802342, 4.97240062320504242813866530914, 5.94567037741323216314380736925, 6.66650127388659503924260990196, 7.81520521953101285156699409397, 8.242141702300957214717329453649, 8.683566104238731047272374834983

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