Properties

Label 2-1850-1.1-c1-0-38
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 2.67·3-s + 4-s − 2.67·6-s − 3.28·7-s + 8-s + 4.15·9-s + 4.83·11-s − 2.67·12-s − 5.67·13-s − 3.28·14-s + 16-s + 5.63·17-s + 4.15·18-s + 4.76·19-s + 8.79·21-s + 4.83·22-s − 2.38·23-s − 2.67·24-s − 5.67·26-s − 3.09·27-s − 3.28·28-s − 7.92·29-s − 7.44·31-s + 32-s − 12.9·33-s + 5.63·34-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.54·3-s + 0.5·4-s − 1.09·6-s − 1.24·7-s + 0.353·8-s + 1.38·9-s + 1.45·11-s − 0.772·12-s − 1.57·13-s − 0.878·14-s + 0.250·16-s + 1.36·17-s + 0.979·18-s + 1.09·19-s + 1.91·21-s + 1.03·22-s − 0.497·23-s − 0.546·24-s − 1.11·26-s − 0.595·27-s − 0.621·28-s − 1.47·29-s − 1.33·31-s + 0.176·32-s − 2.24·33-s + 0.966·34-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: \(1\)
Selberg data: \((2,\ 1850,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.67T + 3T^{2} \)
7 \( 1 + 3.28T + 7T^{2} \)
11 \( 1 - 4.83T + 11T^{2} \)
13 \( 1 + 5.67T + 13T^{2} \)
17 \( 1 - 5.63T + 17T^{2} \)
19 \( 1 - 4.76T + 19T^{2} \)
23 \( 1 + 2.38T + 23T^{2} \)
29 \( 1 + 7.92T + 29T^{2} \)
31 \( 1 + 7.44T + 31T^{2} \)
41 \( 1 + 10.3T + 41T^{2} \)
43 \( 1 - 11.4T + 43T^{2} \)
47 \( 1 - 5.92T + 47T^{2} \)
53 \( 1 + 9.53T + 53T^{2} \)
59 \( 1 + 11.4T + 59T^{2} \)
61 \( 1 + 4.89T + 61T^{2} \)
67 \( 1 + 2.10T + 67T^{2} \)
71 \( 1 + 4.89T + 71T^{2} \)
73 \( 1 + 10.1T + 73T^{2} \)
79 \( 1 - 7.61T + 79T^{2} \)
83 \( 1 + 13.9T + 83T^{2} \)
89 \( 1 - 4.44T + 89T^{2} \)
97 \( 1 + 5.35T + 97T^{2} \)
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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

−9.324683617402783802233033301556, −7.41548929314275333717262350046, −7.17871171923660310165916318813, −6.14645281315181307945576453784, −5.73135121159042644826580624446, −4.95422430234376365463207186813, −3.92460254527544979373050496392, −3.12046170213099853606966350508, −1.48290435010814020346951263517, 0, 1.48290435010814020346951263517, 3.12046170213099853606966350508, 3.92460254527544979373050496392, 4.95422430234376365463207186813, 5.73135121159042644826580624446, 6.14645281315181307945576453784, 7.17871171923660310165916318813, 7.41548929314275333717262350046, 9.324683617402783802233033301556

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