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Properties

Label 2-2208-184.91-c1-0-39
Degree $2$
Conductor $2208$
Sign $0.748 + 0.662i$
Analytic cond. $17.6309$
Root an. cond. $4.19892$
Motivic weight $1$
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 + 2.80·5-s + 0.415·7-s + 9-s − 5.54i·11-s − 3.19i·13-s + 2.80·15-s + 2.01i·17-s + 2.50i·19-s + 0.415·21-s + (2.87 + 3.83i)23-s + 2.86·25-s + 27-s − 4.69i·29-s − 8.16i·31-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.25·5-s + 0.156·7-s + 0.333·9-s − 1.67i·11-s − 0.884i·13-s + 0.724·15-s + 0.488i·17-s + 0.574i·19-s + 0.0906·21-s + (0.599 + 0.800i)23-s + 0.573·25-s + 0.192·27-s − 0.872i·29-s − 1.46i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.748 + 0.662i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.748 + 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2208\)    =    \(2^{5} \cdot 3 \cdot 23\)
Sign: $0.748 + 0.662i$
Analytic conductor: \(17.6309\)
Root analytic conductor: \(4.19892\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2208} (367, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2208,\ (\ :1/2),\ 0.748 + 0.662i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.867099830\)
\(L(\frac12)\) \(\approx\) \(2.867099830\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
23 \( 1 + (-2.87 - 3.83i)T \)
good5 \( 1 - 2.80T + 5T^{2} \)
7 \( 1 - 0.415T + 7T^{2} \)
11 \( 1 + 5.54iT - 11T^{2} \)
13 \( 1 + 3.19iT - 13T^{2} \)
17 \( 1 - 2.01iT - 17T^{2} \)
19 \( 1 - 2.50iT - 19T^{2} \)
29 \( 1 + 4.69iT - 29T^{2} \)
31 \( 1 + 8.16iT - 31T^{2} \)
37 \( 1 - 3.59T + 37T^{2} \)
41 \( 1 - 2.52T + 41T^{2} \)
43 \( 1 - 1.80iT - 43T^{2} \)
47 \( 1 + 3.58iT - 47T^{2} \)
53 \( 1 + 0.270T + 53T^{2} \)
59 \( 1 - 0.657T + 59T^{2} \)
61 \( 1 + 11.8T + 61T^{2} \)
67 \( 1 - 5.41iT - 67T^{2} \)
71 \( 1 + 6.09iT - 71T^{2} \)
73 \( 1 + 2.82T + 73T^{2} \)
79 \( 1 - 15.0T + 79T^{2} \)
83 \( 1 - 11.1iT - 83T^{2} \)
89 \( 1 - 4.02iT - 89T^{2} \)
97 \( 1 - 14.5iT - 97T^{2} \)
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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

−9.073683632645380194853404619147, −8.101504096351331736857915521609, −7.75512619414947987108147750704, −6.29699609310489675162185248593, −5.93833508465656233246471874661, −5.17859051304149486136020645789, −3.86223905802423755121325587525, −3.04894318424963982010712589516, −2.13068228477626562231735046323, −0.969266727222938873732752529316, 1.52063003419243245528930391623, 2.18844304348095850516493426180, 3.11620462042776853673767352399, 4.62273072165431601759698053018, 4.84455327588966507835859352509, 6.13146252639476733000187978402, 6.93106649175130827073503534349, 7.38568799525720049305930530427, 8.623513707539260698636447006559, 9.232555857058941211499333211330

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