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Properties

Label 2-350-5.4-c5-0-11
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 + 3i·3-s − 16·4-s + 12·6-s + 49i·7-s + 64i·8-s + 234·9-s + 405·11-s − 48i·12-s + 391i·13-s + 196·14-s + 256·16-s + 999i·17-s − 936i·18-s − 2.34e3·19-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.192i·3-s − 0.5·4-s + 0.136·6-s + 0.377i·7-s + 0.353i·8-s + 0.962·9-s + 1.00·11-s − 0.0962i·12-s + 0.641i·13-s + 0.267·14-s + 0.250·16-s + 0.838i·17-s − 0.680i·18-s − 1.48·19-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

\(L(3)\) \(\approx\) \(1.551456411\)
\(L(\frac12)\) \(\approx\) \(1.551456411\)
\(L(\frac{7}{2})\) 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 \( 1 - 3iT - 243T^{2} \)
11 \( 1 - 405T + 1.61e5T^{2} \)
13 \( 1 - 391iT - 3.71e5T^{2} \)
17 \( 1 - 999iT - 1.41e6T^{2} \)
19 \( 1 + 2.34e3T + 2.47e6T^{2} \)
23 \( 1 + 2.43e3iT - 6.43e6T^{2} \)
29 \( 1 + 8.25e3T + 2.05e7T^{2} \)
31 \( 1 - 4.01e3T + 2.86e7T^{2} \)
37 \( 1 + 7.04e3iT - 6.93e7T^{2} \)
41 \( 1 - 3.33e3T + 1.15e8T^{2} \)
43 \( 1 - 2.35e4iT - 1.47e8T^{2} \)
47 \( 1 - 1.03e4iT - 2.29e8T^{2} \)
53 \( 1 + 3.08e3iT - 4.18e8T^{2} \)
59 \( 1 - 1.88e4T + 7.14e8T^{2} \)
61 \( 1 - 2.16e4T + 8.44e8T^{2} \)
67 \( 1 - 5.21e4iT - 1.35e9T^{2} \)
71 \( 1 + 2.85e4T + 1.80e9T^{2} \)
73 \( 1 - 7.03e4iT - 2.07e9T^{2} \)
79 \( 1 + 5.88e4T + 3.07e9T^{2} \)
83 \( 1 + 756iT - 3.93e9T^{2} \)
89 \( 1 + 1.35e5T + 5.58e9T^{2} \)
97 \( 1 - 1.10e5iT - 8.58e9T^{2} \)
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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

−10.87524964580864360048042156699, −9.924204737832344596010864951707, −9.147879500394021129993785859046, −8.307446878714609250873939286612, −6.92158222362119014219056018762, −5.97913053468422386248293348860, −4.41217257539128596325672669537, −3.93556482073802739511391746779, −2.28889899083210710493360341396, −1.29871180177463202266503025227, 0.41671397945096206055179035081, 1.74760237765255382078927926585, 3.61827897192229791230810929295, 4.52223876740094904572226963669, 5.75913212469939274370912961172, 6.84495175321044538955280963000, 7.41624060770527472970787346996, 8.530502277944782146789754948484, 9.500413462088595009244174372353, 10.30746709885766710239771793117

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