Properties

Label 2-3850-1.1-c1-0-31
Degree $2$
Conductor $3850$
Sign $1$
Analytic cond. $30.7424$
Root an. cond. $5.54458$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s − 7-s + 8-s − 3·9-s − 11-s + 6·13-s − 14-s + 16-s + 2·17-s − 3·18-s − 4·19-s − 22-s + 4·23-s + 6·26-s − 28-s + 6·29-s + 32-s + 2·34-s − 3·36-s + 2·37-s − 4·38-s − 6·41-s + 4·43-s − 44-s + 4·46-s − 4·47-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s − 9-s − 0.301·11-s + 1.66·13-s − 0.267·14-s + 1/4·16-s + 0.485·17-s − 0.707·18-s − 0.917·19-s − 0.213·22-s + 0.834·23-s + 1.17·26-s − 0.188·28-s + 1.11·29-s + 0.176·32-s + 0.342·34-s − 1/2·36-s + 0.328·37-s − 0.648·38-s − 0.937·41-s + 0.609·43-s − 0.150·44-s + 0.589·46-s − 0.583·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3850\)    =    \(2 \cdot 5^{2} \cdot 7 \cdot 11\)
Sign: $1$
Analytic conductor: \(30.7424\)
Root analytic conductor: \(5.54458\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{3850} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3850,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.840862718\)
\(L(\frac12)\) \(\approx\) \(2.840862718\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 + T \)
11 \( 1 + T \)
good3 \( 1 + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−8.527044098356721696869996324504, −7.79139251965102174785456011339, −6.68527848715894114881498521882, −6.24105337498979840969416095552, −5.54385508956341705884081923791, −4.75563880056112408272227374276, −3.71728152784183348671977379261, −3.16684972416390811064155896405, −2.23600489601986428512099858026, −0.883283840484406346937518621516, 0.883283840484406346937518621516, 2.23600489601986428512099858026, 3.16684972416390811064155896405, 3.71728152784183348671977379261, 4.75563880056112408272227374276, 5.54385508956341705884081923791, 6.24105337498979840969416095552, 6.68527848715894114881498521882, 7.79139251965102174785456011339, 8.527044098356721696869996324504

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