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Properties

Label 2-3328-13.5-c0-0-2
Degree $2$
Conductor $3328$
Sign $0.881 + 0.471i$
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  − 3-s + (0.707 − 0.707i)5-s + (−0.707 − 0.707i)7-s + (1 + i)11-s + (0.707 − 0.707i)13-s + (−0.707 + 0.707i)15-s + i·17-s + (0.707 + 0.707i)21-s + 1.41i·23-s + 27-s + 1.41·29-s + (−1 − i)33-s − 1.00·35-s + (−0.707 − 0.707i)37-s + (−0.707 + 0.707i)39-s + ⋯
L(s)  = 1  − 3-s + (0.707 − 0.707i)5-s + (−0.707 − 0.707i)7-s + (1 + i)11-s + (0.707 − 0.707i)13-s + (−0.707 + 0.707i)15-s + i·17-s + (0.707 + 0.707i)21-s + 1.41i·23-s + 27-s + 1.41·29-s + (−1 − i)33-s − 1.00·35-s + (−0.707 − 0.707i)37-s + (−0.707 + 0.707i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 + 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 + 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3328\)    =    \(2^{8} \cdot 13\)
Sign: $0.881 + 0.471i$
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: \((2,\ 3328,\ (\ :0),\ 0.881 + 0.471i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9872372045\)
\(L(\frac12)\) \(\approx\) \(0.9872372045\)
\(L(1)\) 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 \( 1 + (-0.707 + 0.707i)T \)
good3 \( 1 + T + T^{2} \)
5 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
7 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
11 \( 1 + (-1 - i)T + iT^{2} \)
17 \( 1 - iT - T^{2} \)
19 \( 1 - iT^{2} \)
23 \( 1 - 1.41iT - T^{2} \)
29 \( 1 - 1.41T + T^{2} \)
31 \( 1 - iT^{2} \)
37 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
41 \( 1 - iT^{2} \)
43 \( 1 + iT - T^{2} \)
47 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
53 \( 1 - 1.41T + T^{2} \)
59 \( 1 + iT^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + (-1 + i)T - iT^{2} \)
71 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 - 1.41T + T^{2} \)
83 \( 1 + (1 - i)T - iT^{2} \)
89 \( 1 + iT^{2} \)
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

−8.861702980989496412878057377392, −8.054274874884558530814316687679, −6.88791339298205043924876851959, −6.54797353259923222724864478329, −5.61994927179897205010427293936, −5.21978099986299939232585636297, −4.10589036630767346465375891242, −3.43724585778334133777786594958, −1.84555120008892771865255644502, −0.924966298934209022985765984865, 0.991848729997494632635025186410, 2.52894735991835085369705734373, 3.13049275423748141571399644721, 4.33137426971137953149802076577, 5.29032041527772161086589480255, 6.13750132679705129456962501401, 6.45117273500322342653135578035, 6.81383517736964235035544552705, 8.408549071885945895721114782041, 8.831626350219996984588761744164

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