Properties

Label 2-1850-1.1-c1-0-1
Degree $2$
Conductor $1850$
Sign $1$
Analytic cond. $14.7723$
Root an. cond. $3.84347$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 2.44·3-s + 4-s + 2.44·6-s + 0.449·7-s − 8-s + 2.99·9-s − 4.89·11-s − 2.44·12-s − 4·13-s − 0.449·14-s + 16-s − 4.89·17-s − 2.99·18-s + 8.44·19-s − 1.10·21-s + 4.89·22-s − 0.898·23-s + 2.44·24-s + 4·26-s + 0.449·28-s − 6.44·31-s − 32-s + 11.9·33-s + 4.89·34-s + 2.99·36-s − 37-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.41·3-s + 0.5·4-s + 0.999·6-s + 0.169·7-s − 0.353·8-s + 0.999·9-s − 1.47·11-s − 0.707·12-s − 1.10·13-s − 0.120·14-s + 0.250·16-s − 1.18·17-s − 0.707·18-s + 1.93·19-s − 0.240·21-s + 1.04·22-s − 0.187·23-s + 0.499·24-s + 0.784·26-s + 0.0849·28-s − 1.15·31-s − 0.176·32-s + 2.08·33-s + 0.840·34-s + 0.499·36-s − 0.164·37-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(0.3662497197\)
\(L(\frac12)\) \(\approx\) \(0.3662497197\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
5 \( 1 \)
37 \( 1 + T \)
good3 \( 1 + 2.44T + 3T^{2} \)
7 \( 1 - 0.449T + 7T^{2} \)
11 \( 1 + 4.89T + 11T^{2} \)
13 \( 1 + 4T + 13T^{2} \)
17 \( 1 + 4.89T + 17T^{2} \)
19 \( 1 - 8.44T + 19T^{2} \)
23 \( 1 + 0.898T + 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + 6.44T + 31T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 - 0.449T + 47T^{2} \)
53 \( 1 + 7.79T + 53T^{2} \)
59 \( 1 + 8.44T + 59T^{2} \)
61 \( 1 - 12T + 61T^{2} \)
67 \( 1 - 10.4T + 67T^{2} \)
71 \( 1 + 4.89T + 71T^{2} \)
73 \( 1 + 4T + 73T^{2} \)
79 \( 1 - 1.55T + 79T^{2} \)
83 \( 1 - 14.4T + 83T^{2} \)
89 \( 1 + 3.79T + 89T^{2} \)
97 \( 1 - 2T + 97T^{2} \)
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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

−9.539634505062407237753580038980, −8.382999496451316062056661948183, −7.50247815277804967173666482971, −7.04076746622784735945254544611, −6.00369316019207494121921184293, −5.21832779903434858025545451943, −4.77768532184350806743639605734, −3.12834078045925268178643780899, −1.99164304776830785379259532927, −0.46755908553593451091903809778, 0.46755908553593451091903809778, 1.99164304776830785379259532927, 3.12834078045925268178643780899, 4.77768532184350806743639605734, 5.21832779903434858025545451943, 6.00369316019207494121921184293, 7.04076746622784735945254544611, 7.50247815277804967173666482971, 8.382999496451316062056661948183, 9.539634505062407237753580038980

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