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Properties

Label 2-3328-13.5-c0-0-1
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

\(L(\frac{1}{2})\) \(\approx\) \(0.5629618460\)
\(L(\frac12)\) \(\approx\) \(0.5629618460\)
\(L(1)\) 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

−9.093787337930712075668653426599, −8.348988247066210618850893712547, −7.48909284348875217223459143688, −6.78147900624341803508257659904, −6.23419578781427181814755604812, −5.32776916692961839949951128195, −4.53230174590484081387741843906, −3.90134934504007201679611440278, −2.58238175967000925342056576699, −1.63651564532410532231685125454, 0.41622736699914386261097607715, 1.34855798225777769390683215577, 3.03683872640545921137125659944, 4.00832217688096604356692928227, 4.70825005235869427812366605484, 5.46646347498330350961052526113, 6.01322560834684572460328343623, 7.17736884584133150148216707880, 7.63855000291095691526821660976, 8.429742602217497086179221945197

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