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Properties

Label 2-3328-104.51-c0-0-0
Degree $2$
Conductor $3328$
Sign $-0.707 - 0.707i$
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  − 9-s + i·13-s − 2·17-s − 25-s + 2i·29-s − 49-s + 2i·53-s − 2i·61-s + 81-s + 2i·101-s − 2·113-s − i·117-s + ⋯
L(s)  = 1  − 9-s + i·13-s − 2·17-s − 25-s + 2i·29-s − 49-s + 2i·53-s − 2i·61-s + 81-s + 2i·101-s − 2·113-s − i·117-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3328\)    =    \(2^{8} \cdot 13\)
Sign: $-0.707 - 0.707i$
Analytic conductor: \(1.66088\)
Root analytic conductor: \(1.28875\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3328} (1663, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3328,\ (\ :0),\ -0.707 - 0.707i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4816796887\)
\(L(\frac12)\) \(\approx\) \(0.4816796887\)
\(L(1)\) 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 \( 1 - iT \)
good3 \( 1 + T^{2} \)
5 \( 1 + T^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 - T^{2} \)
17 \( 1 + 2T + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - 2iT - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 - 2iT - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + 2iT - T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - T^{2} \)
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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

−9.029465383546035871878534871576, −8.517702324443327077371987257980, −7.60950684370455604929577867892, −6.69691944472228119945698340349, −6.26473184459110417134015212680, −5.23134159907077580555627241009, −4.51606685992743068737542316678, −3.61552881831126775068940643813, −2.59471942786917437514595359330, −1.71886264181862685897032477003, 0.26082158826684361493157156579, 2.07058571577271981026323014024, 2.79239826263953784973815100106, 3.84876431661759209656499351897, 4.66676577545399729562923295329, 5.62972990018220083024773909531, 6.17531915908915402356208646952, 6.99871129430336670937698577266, 8.025373830392692634039911249747, 8.380697759020509924545190532457

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