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Properties

Label 2-5850-5.4-c1-0-67
Degree $2$
Conductor $5850$
Sign $0.894 + 0.447i$
Analytic cond. $46.7124$
Root an. cond. $6.83465$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·2-s − 4-s − 5i·7-s − i·8-s + 3·11-s + i·13-s + 5·14-s + 16-s + 3i·17-s + 4·19-s + 3i·22-s − 6i·23-s − 26-s + 5i·28-s + 9·29-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s − 1.88i·7-s − 0.353i·8-s + 0.904·11-s + 0.277i·13-s + 1.33·14-s + 0.250·16-s + 0.727i·17-s + 0.917·19-s + 0.639i·22-s − 1.25i·23-s − 0.196·26-s + 0.944i·28-s + 1.67·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5850\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 13\)
Sign: $0.894 + 0.447i$
Analytic conductor: \(46.7124\)
Root analytic conductor: \(6.83465\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5850} (5149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5850,\ (\ :1/2),\ 0.894 + 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.027625292\)
\(L(\frac12)\) \(\approx\) \(2.027625292\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 \)
5 \( 1 \)
13 \( 1 - iT \)
good7 \( 1 + 5iT - 7T^{2} \)
11 \( 1 - 3T + 11T^{2} \)
17 \( 1 - 3iT - 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 + 6iT - 23T^{2} \)
29 \( 1 - 9T + 29T^{2} \)
31 \( 1 - 5T + 31T^{2} \)
37 \( 1 + 2iT - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 2iT - 43T^{2} \)
47 \( 1 + 9iT - 47T^{2} \)
53 \( 1 - 9iT - 53T^{2} \)
59 \( 1 + 9T + 59T^{2} \)
61 \( 1 + T + 61T^{2} \)
67 \( 1 + 5iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 14iT - 73T^{2} \)
79 \( 1 - 16T + 79T^{2} \)
83 \( 1 - 15iT - 83T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 + 8iT - 97T^{2} \)
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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

−7.989441585400118982009954962358, −7.22709627955859053005018759586, −6.63345400432238205142595429399, −6.27942319834309992364544686522, −5.06810941158377703294819402288, −4.30745877570547816007435966240, −3.95501478890688052316325447945, −2.95424717823972850254290737686, −1.39021069772886326040703868159, −0.65787606373635275209591632574, 1.03613172181913335854310145249, 2.00586297407569526267436232887, 2.92203061884521780410050798622, 3.34386277593058466746112364740, 4.64487834160716412977964386495, 5.15056508179130536814559404148, 5.95603111496467144347020744148, 6.53714679799580387883892891291, 7.64858056663539622694664322680, 8.326125621414466357425643526662

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