Properties

Label 2-1850-37.36-c1-0-14
Degree $2$
Conductor $1850$
Sign $0.651 + 0.758i$
Analytic cond. $14.7723$
Root an. cond. $3.84347$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  i·2-s − 0.987·3-s − 4-s + 0.987i·6-s − 4.78·7-s + i·8-s − 2.02·9-s − 5.98·11-s + 0.987·12-s + 3.49i·13-s + 4.78i·14-s + 16-s + 4.96i·17-s + 2.02i·18-s − 7.33i·19-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.570·3-s − 0.5·4-s + 0.403i·6-s − 1.81·7-s + 0.353i·8-s − 0.674·9-s − 1.80·11-s + 0.285·12-s + 0.969i·13-s + 1.28i·14-s + 0.250·16-s + 1.20i·17-s + 0.477i·18-s − 1.68i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.651 + 0.758i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.651 + 0.758i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1850\)    =    \(2 \cdot 5^{2} \cdot 37\)
Sign: $0.651 + 0.758i$
Analytic conductor: \(14.7723\)
Root analytic conductor: \(3.84347\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1850} (1701, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1850,\ (\ :1/2),\ 0.651 + 0.758i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3918000827\)
\(L(\frac12)\) \(\approx\) \(0.3918000827\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
5 \( 1 \)
37 \( 1 + (-4.61 + 3.96i)T \)
good3 \( 1 + 0.987T + 3T^{2} \)
7 \( 1 + 4.78T + 7T^{2} \)
11 \( 1 + 5.98T + 11T^{2} \)
13 \( 1 - 3.49iT - 13T^{2} \)
17 \( 1 - 4.96iT - 17T^{2} \)
19 \( 1 + 7.33iT - 19T^{2} \)
23 \( 1 - 1.74iT - 23T^{2} \)
29 \( 1 + 7.85iT - 29T^{2} \)
31 \( 1 - 3.24iT - 31T^{2} \)
41 \( 1 + 0.530T + 41T^{2} \)
43 \( 1 + 1.76iT - 43T^{2} \)
47 \( 1 + 4.30T + 47T^{2} \)
53 \( 1 - 3.66T + 53T^{2} \)
59 \( 1 - 2.15iT - 59T^{2} \)
61 \( 1 + 3.06iT - 61T^{2} \)
67 \( 1 + 3.79T + 67T^{2} \)
71 \( 1 + 8.47T + 71T^{2} \)
73 \( 1 + 9.05T + 73T^{2} \)
79 \( 1 - 5.56iT - 79T^{2} \)
83 \( 1 - 3.77T + 83T^{2} \)
89 \( 1 - 8.45iT - 89T^{2} \)
97 \( 1 - 3.64iT - 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.272974033153659583984807201972, −8.613429322454512032450201428995, −7.55065036994359134158429767511, −6.54796714694385637354638918508, −5.92935792594687912575293890331, −5.08696653845092171583962538529, −4.04468529673381415467890616011, −2.97226492941450876084050160202, −2.39601615964950228877261919909, −0.39140680920050765887373076481, 0.41752724861181094694911006561, 2.84089289920655051067834980325, 3.25642334741473867246300037544, 4.76535812052473793828083797030, 5.69099404908836035773261315606, 5.90525255105979795470077446867, 6.92483169565324421484741786637, 7.69474423796452817807163308051, 8.424019051268053418249175410046, 9.367592682427487130905614545786

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