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Properties

Label 2-350-5.4-c5-0-19
Degree $2$
Conductor $350$
Sign $0.447 - 0.894i$
Analytic cond. $56.1343$
Root an. cond. $7.49228$
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  − 4i·2-s + 23i·3-s − 16·4-s + 92·6-s + 49i·7-s + 64i·8-s − 286·9-s + 555·11-s − 368i·12-s + 241i·13-s + 196·14-s + 256·16-s − 1.49e3i·17-s + 1.14e3i·18-s + 2.03e3·19-s + ⋯
L(s)  = 1  − 0.707i·2-s + 1.47i·3-s − 0.5·4-s + 1.04·6-s + 0.377i·7-s + 0.353i·8-s − 1.17·9-s + 1.38·11-s − 0.737i·12-s + 0.395i·13-s + 0.267·14-s + 0.250·16-s − 1.25i·17-s + 0.832i·18-s + 1.29·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $0.447 - 0.894i$
Analytic conductor: \(56.1343\)
Root analytic conductor: \(7.49228\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{350} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 350,\ (\ :5/2),\ 0.447 - 0.894i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.210828236\)
\(L(\frac12)\) \(\approx\) \(2.210828236\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 4iT \)
5 \( 1 \)
7 \( 1 - 49iT \)
good3 \( 1 - 23iT - 243T^{2} \)
11 \( 1 - 555T + 1.61e5T^{2} \)
13 \( 1 - 241iT - 3.71e5T^{2} \)
17 \( 1 + 1.49e3iT - 1.41e6T^{2} \)
19 \( 1 - 2.03e3T + 2.47e6T^{2} \)
23 \( 1 - 1.23e3iT - 6.43e6T^{2} \)
29 \( 1 - 5.00e3T + 2.05e7T^{2} \)
31 \( 1 - 5.69e3T + 2.86e7T^{2} \)
37 \( 1 + 5.60e3iT - 6.93e7T^{2} \)
41 \( 1 + 2.42e3T + 1.15e8T^{2} \)
43 \( 1 + 602iT - 1.47e8T^{2} \)
47 \( 1 + 2.31e4iT - 2.29e8T^{2} \)
53 \( 1 - 2.52e4iT - 4.18e8T^{2} \)
59 \( 1 + 5.72e3T + 7.14e8T^{2} \)
61 \( 1 + 3.61e4T + 8.44e8T^{2} \)
67 \( 1 - 6.61e4iT - 1.35e9T^{2} \)
71 \( 1 - 1.60e4T + 1.80e9T^{2} \)
73 \( 1 - 8.04e4iT - 2.07e9T^{2} \)
79 \( 1 - 6.41e4T + 3.07e9T^{2} \)
83 \( 1 - 1.06e5iT - 3.93e9T^{2} \)
89 \( 1 - 7.16e4T + 5.58e9T^{2} \)
97 \( 1 - 1.51e5iT - 8.58e9T^{2} \)
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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

−10.79436551992783854040129524485, −9.665584866555970450980743718289, −9.456745655161154827255724815166, −8.531504222638364019542957352750, −6.96674019568112547110125440797, −5.53427091592031866883421927762, −4.63320630788029861859570397367, −3.75633852047851529771488045172, −2.77116584728048343214059821393, −1.06191224531082288781426379242, 0.73413857434771612288531027909, 1.55533415445948253457156174616, 3.28007492211831578059512920654, 4.66975685954181328755070924381, 6.25026670490442218478676980501, 6.49643343210158103303950850906, 7.63559629086931441032039897191, 8.221087832401596413136055544246, 9.301828437467604673834132144747, 10.46105753580863299980992283273

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