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Properties

Label 2-350-5.4-c5-0-31
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

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

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