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Properties

Label 2-354-177.176-c3-0-58
Degree $2$
Conductor $354$
Sign $-0.995 - 0.0985i$
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 + (−1.46 − 4.98i)3-s + 4·4-s − 18.9i·5-s + (−2.93 − 9.96i)6-s − 10.1·7-s + 8·8-s + (−22.6 + 14.6i)9-s − 37.9i·10-s + 10.1·11-s + (−5.86 − 19.9i)12-s − 82.2i·13-s − 20.3·14-s + (−94.6 + 27.8i)15-s + 16·16-s + 92.4i·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.282 − 0.959i)3-s + 0.5·4-s − 1.69i·5-s + (−0.199 − 0.678i)6-s − 0.550·7-s + 0.353·8-s + (−0.840 + 0.541i)9-s − 1.20i·10-s + 0.278·11-s + (−0.141 − 0.479i)12-s − 1.75i·13-s − 0.389·14-s + (−1.62 + 0.479i)15-s + 0.250·16-s + 1.31i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0985i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.995 - 0.0985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(354\)    =    \(2 \cdot 3 \cdot 59\)
Sign: $-0.995 - 0.0985i$
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: \((2,\ 354,\ (\ :3/2),\ -0.995 - 0.0985i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.787651472\)
\(L(\frac12)\) \(\approx\) \(1.787651472\)
\(L(\frac{5}{2})\) 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 \( 1 + (1.46 + 4.98i)T \)
59 \( 1 + (170. - 420. i)T \)
good5 \( 1 + 18.9iT - 125T^{2} \)
7 \( 1 + 10.1T + 343T^{2} \)
11 \( 1 - 10.1T + 1.33e3T^{2} \)
13 \( 1 + 82.2iT - 2.19e3T^{2} \)
17 \( 1 - 92.4iT - 4.91e3T^{2} \)
19 \( 1 - 23.3T + 6.85e3T^{2} \)
23 \( 1 + 39.6T + 1.21e4T^{2} \)
29 \( 1 + 118. iT - 2.43e4T^{2} \)
31 \( 1 - 30.7iT - 2.97e4T^{2} \)
37 \( 1 - 221. iT - 5.06e4T^{2} \)
41 \( 1 - 63.0iT - 6.89e4T^{2} \)
43 \( 1 - 147. iT - 7.95e4T^{2} \)
47 \( 1 - 37.8T + 1.03e5T^{2} \)
53 \( 1 + 409. iT - 1.48e5T^{2} \)
61 \( 1 + 709. iT - 2.26e5T^{2} \)
67 \( 1 + 1.04e3iT - 3.00e5T^{2} \)
71 \( 1 + 361. iT - 3.57e5T^{2} \)
73 \( 1 + 327. iT - 3.89e5T^{2} \)
79 \( 1 + 238.T + 4.93e5T^{2} \)
83 \( 1 - 156.T + 5.71e5T^{2} \)
89 \( 1 - 1.09e3T + 7.04e5T^{2} \)
97 \( 1 - 1.33e3iT - 9.12e5T^{2} \)
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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

−10.78521696646517998236503255423, −9.637792560461850531298869628467, −8.290379817983711031486974315805, −7.912347715598585678165994338006, −6.35994729300013645494913050421, −5.67295013826337322114590459146, −4.76069501866570997901963804815, −3.32052886519233372547396854443, −1.66067733548263024930904056268, −0.49699495062215995708697673426, 2.48636167699635804629060217815, 3.45494511230808009712990981538, 4.34355011993229899431574248160, 5.70777935681613474035657637998, 6.67419348748684145921231347750, 7.20903179833426419415865612366, 9.100583691049961074917467891950, 9.849103568752875940904441505081, 10.76052539465284004091236948013, 11.48310161386867278794698940443

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