Properties

Label 2-354-177.176-c5-0-59
Degree $2$
Conductor $354$
Sign $-0.0765 + 0.997i$
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 + (1.66 + 15.4i)3-s + 16·4-s − 65.2i·5-s + (−6.65 − 61.9i)6-s − 43.9·7-s − 64·8-s + (−237. + 51.5i)9-s + 261. i·10-s − 184.·11-s + (26.6 + 247. i)12-s + 483. i·13-s + 175.·14-s + (1.01e3 − 108. i)15-s + 256·16-s + 1.66e3i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.106 + 0.994i)3-s + 0.5·4-s − 1.16i·5-s + (−0.0754 − 0.703i)6-s − 0.339·7-s − 0.353·8-s + (−0.977 + 0.212i)9-s + 0.825i·10-s − 0.459·11-s + (0.0533 + 0.497i)12-s + 0.792i·13-s + 0.239·14-s + (1.16 − 0.124i)15-s + 0.250·16-s + 1.39i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0765 + 0.997i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.0765 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(354\)    =    \(2 \cdot 3 \cdot 59\)
Sign: $-0.0765 + 0.997i$
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: \((2,\ 354,\ (\ :5/2),\ -0.0765 + 0.997i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.4939275139\)
\(L(\frac12)\) \(\approx\) \(0.4939275139\)
\(L(\frac{7}{2})\) 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 \( 1 + (-1.66 - 15.4i)T \)
59 \( 1 + (2.62e4 + 4.88e3i)T \)
good5 \( 1 + 65.2iT - 3.12e3T^{2} \)
7 \( 1 + 43.9T + 1.68e4T^{2} \)
11 \( 1 + 184.T + 1.61e5T^{2} \)
13 \( 1 - 483. iT - 3.71e5T^{2} \)
17 \( 1 - 1.66e3iT - 1.41e6T^{2} \)
19 \( 1 - 464.T + 2.47e6T^{2} \)
23 \( 1 - 2.98e3T + 6.43e6T^{2} \)
29 \( 1 + 6.84e3iT - 2.05e7T^{2} \)
31 \( 1 - 9.47e3iT - 2.86e7T^{2} \)
37 \( 1 - 4.96e3iT - 6.93e7T^{2} \)
41 \( 1 + 1.88e4iT - 1.15e8T^{2} \)
43 \( 1 - 3.62e3iT - 1.47e8T^{2} \)
47 \( 1 + 7.49e3T + 2.29e8T^{2} \)
53 \( 1 + 8.88e3iT - 4.18e8T^{2} \)
61 \( 1 + 3.76e4iT - 8.44e8T^{2} \)
67 \( 1 - 2.94e4iT - 1.35e9T^{2} \)
71 \( 1 + 4.55e4iT - 1.80e9T^{2} \)
73 \( 1 + 7.20e4iT - 2.07e9T^{2} \)
79 \( 1 + 3.62e4T + 3.07e9T^{2} \)
83 \( 1 + 2.57e3T + 3.93e9T^{2} \)
89 \( 1 + 7.28e4T + 5.58e9T^{2} \)
97 \( 1 + 3.00e4iT - 8.58e9T^{2} \)
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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

−10.24651233830198589392739980592, −9.358391388598759795270877582742, −8.745404359548380721896328426098, −8.015365945088739228003102268405, −6.53538956848268587326738568613, −5.36358869921610467858935493601, −4.44502278251158641422824020445, −3.21520771040132455696300898673, −1.65723072204058247214126457291, −0.17886893602167517158052805099, 1.03027316883371640795882280825, 2.66367439055619432490064341675, 3.06686031803362855964023411171, 5.36182822656451861022167995538, 6.48164397752508315203002087490, 7.22689097799207950054966990270, 7.80717032988482251956492366482, 9.005338089996605906172103781410, 9.942401181419689506329904318795, 10.99820341632534944296200634602

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