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Properties

Label 2-354-177.176-c5-0-88
Degree $2$
Conductor $354$
Sign $-0.961 + 0.275i$
Analytic cond. $56.7758$
Root an. cond. $7.53497$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·2-s + (−3.72 + 15.1i)3-s + 16·4-s − 73.5i·5-s + (14.8 − 60.5i)6-s + 107.·7-s − 64·8-s + (−215. − 112. i)9-s + 294. i·10-s + 36.3·11-s + (−59.5 + 242. i)12-s − 785. i·13-s − 431.·14-s + (1.11e3 + 273. i)15-s + 256·16-s − 1.63e3i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.238 + 0.971i)3-s + 0.5·4-s − 1.31i·5-s + (0.168 − 0.686i)6-s + 0.831·7-s − 0.353·8-s + (−0.885 − 0.463i)9-s + 0.930i·10-s + 0.0905·11-s + (−0.119 + 0.485i)12-s − 1.28i·13-s − 0.588·14-s + (1.27 + 0.314i)15-s + 0.250·16-s − 1.37i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.961 + 0.275i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.961 + 0.275i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(354\)    =    \(2 \cdot 3 \cdot 59\)
Sign: $-0.961 + 0.275i$
Analytic conductor: \(56.7758\)
Root analytic conductor: \(7.53497\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{354} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 354,\ (\ :5/2),\ -0.961 + 0.275i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.4496716727\)
\(L(\frac12)\) \(\approx\) \(0.4496716727\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 4T \)
3 \( 1 + (3.72 - 15.1i)T \)
59 \( 1 + (1.32e4 + 2.32e4i)T \)
good5 \( 1 + 73.5iT - 3.12e3T^{2} \)
7 \( 1 - 107.T + 1.68e4T^{2} \)
11 \( 1 - 36.3T + 1.61e5T^{2} \)
13 \( 1 + 785. iT - 3.71e5T^{2} \)
17 \( 1 + 1.63e3iT - 1.41e6T^{2} \)
19 \( 1 + 509.T + 2.47e6T^{2} \)
23 \( 1 + 2.11e3T + 6.43e6T^{2} \)
29 \( 1 + 5.63e3iT - 2.05e7T^{2} \)
31 \( 1 + 359. iT - 2.86e7T^{2} \)
37 \( 1 - 1.00e4iT - 6.93e7T^{2} \)
41 \( 1 - 1.65e4iT - 1.15e8T^{2} \)
43 \( 1 - 2.31e4iT - 1.47e8T^{2} \)
47 \( 1 + 2.92e3T + 2.29e8T^{2} \)
53 \( 1 - 4.06e3iT - 4.18e8T^{2} \)
61 \( 1 - 1.71e4iT - 8.44e8T^{2} \)
67 \( 1 + 1.83e4iT - 1.35e9T^{2} \)
71 \( 1 + 5.60e4iT - 1.80e9T^{2} \)
73 \( 1 - 6.09e4iT - 2.07e9T^{2} \)
79 \( 1 - 8.99e4T + 3.07e9T^{2} \)
83 \( 1 + 4.20e4T + 3.93e9T^{2} \)
89 \( 1 + 1.42e5T + 5.58e9T^{2} \)
97 \( 1 + 1.42e5iT - 8.58e9T^{2} \)
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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

−9.927056840857289866224247927288, −9.473244344353251995270357346375, −8.296644722300489532412214012906, −7.977191569650651095115754486868, −6.15871809668483899275366284199, −5.10700058013324847763507335684, −4.47547966202049123200517581209, −2.87570277825366742547562456400, −1.17842195147782322448517892609, −0.15798579933725591655783583689, 1.61669379723312872500944332377, 2.25710106706240802281614058566, 3.82047442931726460970548983968, 5.63423619590802749299670195729, 6.65864062391116276763703958845, 7.17460309890828838507450911519, 8.167949042092031570835006762832, 8.997171744707587006051059784124, 10.53849041094708303661427597445, 10.88312440755202789598572451883

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