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Properties

Label 2-3248-3248.1259-c0-0-0
Degree $2$
Conductor $3248$
Sign $0.201 - 0.979i$
Analytic cond. $1.62096$
Root an. cond. $1.27317$
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  + i·2-s − 4-s − 7-s − i·8-s − 9-s + 2·11-s − i·14-s + 16-s − i·18-s + 2i·22-s − i·25-s + 28-s + i·29-s + i·32-s + 36-s + 2·37-s + ⋯
L(s)  = 1  + i·2-s − 4-s − 7-s − i·8-s − 9-s + 2·11-s − i·14-s + 16-s − i·18-s + 2i·22-s − i·25-s + 28-s + i·29-s + i·32-s + 36-s + 2·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.201 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.201 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3248\)    =    \(2^{4} \cdot 7 \cdot 29\)
Sign: $0.201 - 0.979i$
Analytic conductor: \(1.62096\)
Root analytic conductor: \(1.27317\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3248} (1259, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3248,\ (\ :0),\ 0.201 - 0.979i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9996140016\)
\(L(\frac12)\) \(\approx\) \(0.9996140016\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
7 \( 1 + T \)
29 \( 1 - iT \)
good3 \( 1 + T^{2} \)
5 \( 1 + iT^{2} \)
11 \( 1 - 2T + T^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + T^{2} \)
31 \( 1 + iT^{2} \)
37 \( 1 - 2T + T^{2} \)
41 \( 1 - iT^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + (-1 - i)T + iT^{2} \)
59 \( 1 + iT^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + (-1 + i)T - iT^{2} \)
71 \( 1 - 2iT - T^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 + (1 - i)T - iT^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + iT^{2} \)
97 \( 1 + iT^{2} \)
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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

−8.917921474910340880417822593487, −8.359491086471494577215781495456, −7.34061356570875694463317342982, −6.56059215208243323846002820976, −6.22100396553270048216950933609, −5.47067254707000836751743148612, −4.30516417242503364543832677690, −3.73475825308054763714285040756, −2.74455512367905841630091617610, −0.965233113394577134863110134092, 0.848475870737398988176698540593, 2.13171602887561559556259490452, 3.14137697108325333499900137927, 3.75058086078739729740128362060, 4.51269149323768553180447434665, 5.74654412253157256342506509526, 6.19987342142849872194238143306, 7.15506385365185897838200250574, 8.236280667627223465900401460496, 8.959911730797695258485378241839

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