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Properties

Label 2-350-5.4-c9-0-53
Degree $2$
Conductor $350$
Sign $-0.447 + 0.894i$
Analytic cond. $180.262$
Root an. cond. $13.4261$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 16i·2-s + 6i·3-s − 256·4-s + 96·6-s − 2.40e3i·7-s + 4.09e3i·8-s + 1.96e4·9-s − 5.41e4·11-s − 1.53e3i·12-s + 1.13e5i·13-s − 3.84e4·14-s + 6.55e4·16-s + 6.26e3i·17-s − 3.14e5i·18-s − 2.57e5·19-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.0427i·3-s − 0.5·4-s + 0.0302·6-s − 0.377i·7-s + 0.353i·8-s + 0.998·9-s − 1.11·11-s − 0.0213i·12-s + 1.09i·13-s − 0.267·14-s + 0.250·16-s + 0.0181i·17-s − 0.705i·18-s − 0.452·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $-0.447 + 0.894i$
Analytic conductor: \(180.262\)
Root analytic conductor: \(13.4261\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{350} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 350,\ (\ :9/2),\ -0.447 + 0.894i)\)

Particular Values

\(L(5)\) \(\approx\) \(1.399983985\)
\(L(\frac12)\) \(\approx\) \(1.399983985\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 16iT \)
5 \( 1 \)
7 \( 1 + 2.40e3iT \)
good3 \( 1 - 6iT - 1.96e4T^{2} \)
11 \( 1 + 5.41e4T + 2.35e9T^{2} \)
13 \( 1 - 1.13e5iT - 1.06e10T^{2} \)
17 \( 1 - 6.26e3iT - 1.18e11T^{2} \)
19 \( 1 + 2.57e5T + 3.22e11T^{2} \)
23 \( 1 - 2.66e5iT - 1.80e12T^{2} \)
29 \( 1 + 1.57e6T + 1.45e13T^{2} \)
31 \( 1 + 4.63e6T + 2.64e13T^{2} \)
37 \( 1 + 1.19e7iT - 1.29e14T^{2} \)
41 \( 1 - 2.19e7T + 3.27e14T^{2} \)
43 \( 1 + 2.75e7iT - 5.02e14T^{2} \)
47 \( 1 - 5.29e7iT - 1.11e15T^{2} \)
53 \( 1 + 1.62e7iT - 3.29e15T^{2} \)
59 \( 1 - 1.40e8T + 8.66e15T^{2} \)
61 \( 1 + 2.02e8T + 1.16e16T^{2} \)
67 \( 1 - 1.53e8iT - 2.72e16T^{2} \)
71 \( 1 - 2.79e8T + 4.58e16T^{2} \)
73 \( 1 - 4.04e8iT - 5.88e16T^{2} \)
79 \( 1 - 1.30e8T + 1.19e17T^{2} \)
83 \( 1 + 4.20e8iT - 1.86e17T^{2} \)
89 \( 1 - 4.69e8T + 3.50e17T^{2} \)
97 \( 1 + 8.72e8iT - 7.60e17T^{2} \)
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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

−9.764807218103720229296731663074, −8.983464058543178588248297296024, −7.75262583345732643516394183863, −6.98969692455385470097531493371, −5.62132679107249204971743304004, −4.49130043478465942596555964602, −3.79986182508635989792637364196, −2.42288512393902712457499933659, −1.55933261459968695144387929411, −0.33709623664977323950419807856, 0.77532418404833773005128429999, 2.18067783750437701244104902462, 3.41916047053294819764552670628, 4.68324187592955777701951024613, 5.47420544626599636551367971838, 6.47395637987354186108415060490, 7.57525853145150950891953335014, 8.109295111690129558276533718396, 9.257805859822280889016099180427, 10.18077920213718999642246956195

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