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Properties

Label 2-354-177.176-c3-0-21
Degree $2$
Conductor $354$
Sign $0.800 - 0.599i$
Analytic cond. $20.8866$
Root an. cond. $4.57019$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + (5.16 − 0.541i)3-s + 4·4-s − 1.12i·5-s + (−10.3 + 1.08i)6-s − 8.16·7-s − 8·8-s + (26.4 − 5.59i)9-s + 2.24i·10-s − 24.0·11-s + (20.6 − 2.16i)12-s + 67.9i·13-s + 16.3·14-s + (−0.608 − 5.80i)15-s + 16·16-s + 20.4i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.994 − 0.104i)3-s + 0.5·4-s − 0.100i·5-s + (−0.703 + 0.0736i)6-s − 0.441·7-s − 0.353·8-s + (0.978 − 0.207i)9-s + 0.0710i·10-s − 0.658·11-s + (0.497 − 0.0521i)12-s + 1.45i·13-s + 0.311·14-s + (−0.0104 − 0.0999i)15-s + 0.250·16-s + 0.291i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.800 - 0.599i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.800 - 0.599i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(354\)    =    \(2 \cdot 3 \cdot 59\)
Sign: $0.800 - 0.599i$
Analytic conductor: \(20.8866\)
Root analytic conductor: \(4.57019\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{354} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 354,\ (\ :3/2),\ 0.800 - 0.599i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.804703638\)
\(L(\frac12)\) \(\approx\) \(1.804703638\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 2T \)
3 \( 1 + (-5.16 + 0.541i)T \)
59 \( 1 + (-388. + 232. i)T \)
good5 \( 1 + 1.12iT - 125T^{2} \)
7 \( 1 + 8.16T + 343T^{2} \)
11 \( 1 + 24.0T + 1.33e3T^{2} \)
13 \( 1 - 67.9iT - 2.19e3T^{2} \)
17 \( 1 - 20.4iT - 4.91e3T^{2} \)
19 \( 1 - 108.T + 6.85e3T^{2} \)
23 \( 1 - 50.2T + 1.21e4T^{2} \)
29 \( 1 - 245. iT - 2.43e4T^{2} \)
31 \( 1 - 189. iT - 2.97e4T^{2} \)
37 \( 1 + 29.0iT - 5.06e4T^{2} \)
41 \( 1 + 419. iT - 6.89e4T^{2} \)
43 \( 1 + 434. iT - 7.95e4T^{2} \)
47 \( 1 - 418.T + 1.03e5T^{2} \)
53 \( 1 - 614. iT - 1.48e5T^{2} \)
61 \( 1 + 438. iT - 2.26e5T^{2} \)
67 \( 1 - 971. iT - 3.00e5T^{2} \)
71 \( 1 - 967. iT - 3.57e5T^{2} \)
73 \( 1 - 656. iT - 3.89e5T^{2} \)
79 \( 1 - 185.T + 4.93e5T^{2} \)
83 \( 1 - 218.T + 5.71e5T^{2} \)
89 \( 1 - 385.T + 7.04e5T^{2} \)
97 \( 1 - 388. iT - 9.12e5T^{2} \)
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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

−10.85871973472234685013881317619, −10.03443210273764025944211026512, −9.012925796932246899848524087867, −8.696766817578457838674270813667, −7.27352160758441976915712978733, −6.90971049381829640176805773747, −5.22783782126229014487097529199, −3.71275390293333440931204623380, −2.59896967160349400154651961998, −1.27778385788932092655426626358, 0.797988589016059394234234115762, 2.59256040784995257500157282077, 3.29981868036457437343278939132, 4.98051815478385400508611972514, 6.33228953962089338511760157544, 7.71111267557424042807349377374, 7.916185971907701178692836922209, 9.222515416468935426207850670135, 9.858655621933519576354082966662, 10.59655428750418352024270968101

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