Properties

Label 2-354-177.176-c5-0-19
Degree $2$
Conductor $354$
Sign $-0.329 - 0.944i$
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 + (−9.75 + 12.1i)3-s + 16·4-s − 43.2i·5-s + (−39.0 + 48.6i)6-s − 5.13·7-s + 64·8-s + (−52.5 − 237. i)9-s − 173. i·10-s − 262.·11-s + (−156. + 194. i)12-s − 296. i·13-s − 20.5·14-s + (526. + 422. i)15-s + 256·16-s + 1.33e3i·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.625 + 0.779i)3-s + 0.5·4-s − 0.774i·5-s + (−0.442 + 0.551i)6-s − 0.0396·7-s + 0.353·8-s + (−0.216 − 0.976i)9-s − 0.547i·10-s − 0.654·11-s + (−0.312 + 0.389i)12-s − 0.485i·13-s − 0.0280·14-s + (0.603 + 0.484i)15-s + 0.250·16-s + 1.12i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.329 - 0.944i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.329 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(354\)    =    \(2 \cdot 3 \cdot 59\)
Sign: $-0.329 - 0.944i$
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: \((2,\ 354,\ (\ :5/2),\ -0.329 - 0.944i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.596591824\)
\(L(\frac12)\) \(\approx\) \(1.596591824\)
\(L(\frac{7}{2})\) 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 \( 1 + (9.75 - 12.1i)T \)
59 \( 1 + (1.41e4 - 2.26e4i)T \)
good5 \( 1 + 43.2iT - 3.12e3T^{2} \)
7 \( 1 + 5.13T + 1.68e4T^{2} \)
11 \( 1 + 262.T + 1.61e5T^{2} \)
13 \( 1 + 296. iT - 3.71e5T^{2} \)
17 \( 1 - 1.33e3iT - 1.41e6T^{2} \)
19 \( 1 + 1.34e3T + 2.47e6T^{2} \)
23 \( 1 - 1.85e3T + 6.43e6T^{2} \)
29 \( 1 - 1.12e3iT - 2.05e7T^{2} \)
31 \( 1 - 6.10e3iT - 2.86e7T^{2} \)
37 \( 1 - 6.96e3iT - 6.93e7T^{2} \)
41 \( 1 + 1.64e3iT - 1.15e8T^{2} \)
43 \( 1 + 1.56e3iT - 1.47e8T^{2} \)
47 \( 1 - 1.35e4T + 2.29e8T^{2} \)
53 \( 1 - 1.28e4iT - 4.18e8T^{2} \)
61 \( 1 - 5.07e4iT - 8.44e8T^{2} \)
67 \( 1 + 1.30e3iT - 1.35e9T^{2} \)
71 \( 1 - 1.76e4iT - 1.80e9T^{2} \)
73 \( 1 + 1.63e4iT - 2.07e9T^{2} \)
79 \( 1 - 6.66e4T + 3.07e9T^{2} \)
83 \( 1 + 5.27e4T + 3.93e9T^{2} \)
89 \( 1 + 8.97e4T + 5.58e9T^{2} \)
97 \( 1 - 2.49e4iT - 8.58e9T^{2} \)
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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

−10.71979754505948623450005528809, −10.44897996644539662231818918474, −9.079144471317051822083147084524, −8.239674697997522156672777172237, −6.79342717239334377852074929389, −5.75352674200293449601958491859, −4.99582102760387736858433980489, −4.16204136777757013094330056380, −2.95913699632621623069134222861, −1.17388451750098654382417882162, 0.36057485560799837503960886688, 2.04337672863784059233827808879, 2.95982788329463161356563523108, 4.53476663116384184007003351267, 5.53080860768983133796886785658, 6.56008303653574981965104708478, 7.15371283251132690800524275687, 8.123531400099413425880633891479, 9.631219734071857906553544538819, 10.94158881123371295337709839540

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