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Properties

Label 2-350-5.4-c5-0-17
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 − 8i·3-s − 16·4-s + 32·6-s − 49i·7-s − 64i·8-s + 179·9-s − 340·11-s + 128i·12-s + 294i·13-s + 196·14-s + 256·16-s + 1.22e3i·17-s + 716i·18-s − 2.43e3·19-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.513i·3-s − 0.5·4-s + 0.362·6-s − 0.377i·7-s − 0.353i·8-s + 0.736·9-s − 0.847·11-s + 0.256i·12-s + 0.482i·13-s + 0.267·14-s + 0.250·16-s + 1.02i·17-s + 0.520i·18-s − 1.54·19-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

\(L(3)\) \(\approx\) \(1.687995545\)
\(L(\frac12)\) \(\approx\) \(1.687995545\)
\(L(\frac{7}{2})\) 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 \( 1 + 8iT - 243T^{2} \)
11 \( 1 + 340T + 1.61e5T^{2} \)
13 \( 1 - 294iT - 3.71e5T^{2} \)
17 \( 1 - 1.22e3iT - 1.41e6T^{2} \)
19 \( 1 + 2.43e3T + 2.47e6T^{2} \)
23 \( 1 + 2.00e3iT - 6.43e6T^{2} \)
29 \( 1 - 6.74e3T + 2.05e7T^{2} \)
31 \( 1 - 8.85e3T + 2.86e7T^{2} \)
37 \( 1 - 9.18e3iT - 6.93e7T^{2} \)
41 \( 1 + 1.45e4T + 1.15e8T^{2} \)
43 \( 1 + 8.10e3iT - 1.47e8T^{2} \)
47 \( 1 + 312iT - 2.29e8T^{2} \)
53 \( 1 - 1.46e4iT - 4.18e8T^{2} \)
59 \( 1 - 2.76e4T + 7.14e8T^{2} \)
61 \( 1 - 3.43e4T + 8.44e8T^{2} \)
67 \( 1 - 1.23e4iT - 1.35e9T^{2} \)
71 \( 1 - 3.69e4T + 1.80e9T^{2} \)
73 \( 1 - 6.17e4iT - 2.07e9T^{2} \)
79 \( 1 - 6.47e4T + 3.07e9T^{2} \)
83 \( 1 - 7.70e4iT - 3.93e9T^{2} \)
89 \( 1 - 8.16e3T + 5.58e9T^{2} \)
97 \( 1 - 2.06e4iT - 8.58e9T^{2} \)
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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

−10.45872320305995966690063357141, −10.10601504538735358614367680089, −8.480044338069604170768183440053, −8.107454686399614822643316366378, −6.75419593028593002724795195065, −6.44094173925662194992837631202, −4.88365124353271090771579103576, −4.04920562286277137153920910164, −2.31324394472296144315754806874, −0.897097503485893884329178376324, 0.56353517053865229587913572201, 2.13468701313522578176703887711, 3.20337790692895664389565678350, 4.48260092702335231231270061977, 5.21261292682522882761136799951, 6.60645263205309433371037271602, 7.909458893826201233397615466203, 8.794215056352537391386176721903, 9.915799826425612805473910191424, 10.33486471105591164635077800631

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