Properties

Label 2-3850-1.1-c1-0-22
Degree $2$
Conductor $3850$
Sign $1$
Analytic cond. $30.7424$
Root an. cond. $5.54458$
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 + 1.41·3-s + 4-s − 1.41·6-s + 7-s − 8-s − 0.999·9-s − 11-s + 1.41·12-s − 3.24·13-s − 14-s + 16-s + 2.58·17-s + 0.999·18-s + 0.171·19-s + 1.41·21-s + 22-s + 1.58·23-s − 1.41·24-s + 3.24·26-s − 5.65·27-s + 28-s + 8.65·29-s − 0.171·31-s − 32-s − 1.41·33-s − 2.58·34-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.816·3-s + 0.5·4-s − 0.577·6-s + 0.377·7-s − 0.353·8-s − 0.333·9-s − 0.301·11-s + 0.408·12-s − 0.899·13-s − 0.267·14-s + 0.250·16-s + 0.627·17-s + 0.235·18-s + 0.0393·19-s + 0.308·21-s + 0.213·22-s + 0.330·23-s − 0.288·24-s + 0.635·26-s − 1.08·27-s + 0.188·28-s + 1.60·29-s − 0.0308·31-s − 0.176·32-s − 0.246·33-s − 0.443·34-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: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3850,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.687273123\)
\(L(\frac12)\) \(\approx\) \(1.687273123\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.41T + 3T^{2} \)
13 \( 1 + 3.24T + 13T^{2} \)
17 \( 1 - 2.58T + 17T^{2} \)
19 \( 1 - 0.171T + 19T^{2} \)
23 \( 1 - 1.58T + 23T^{2} \)
29 \( 1 - 8.65T + 29T^{2} \)
31 \( 1 + 0.171T + 31T^{2} \)
37 \( 1 - 2.82T + 37T^{2} \)
41 \( 1 + 6.82T + 41T^{2} \)
43 \( 1 - 10.8T + 43T^{2} \)
47 \( 1 + 6.48T + 47T^{2} \)
53 \( 1 - 12.2T + 53T^{2} \)
59 \( 1 - 4T + 59T^{2} \)
61 \( 1 + 12.8T + 61T^{2} \)
67 \( 1 - 3.41T + 67T^{2} \)
71 \( 1 - 10.0T + 71T^{2} \)
73 \( 1 - 15.3T + 73T^{2} \)
79 \( 1 - 11.0T + 79T^{2} \)
83 \( 1 - 13.8T + 83T^{2} \)
89 \( 1 + 6.89T + 89T^{2} \)
97 \( 1 - 2.07T + 97T^{2} \)
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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

−8.352440006252109968846501658767, −7.988458870531575125301197797382, −7.29887125480504178640242915193, −6.47373761562333669165555384503, −5.49541321575274865302432080349, −4.76034866808911937907595078316, −3.58992805809914000026707309915, −2.74516375363191670439954034158, −2.12224228292733447890301059755, −0.791132535421916844230653318946, 0.791132535421916844230653318946, 2.12224228292733447890301059755, 2.74516375363191670439954034158, 3.58992805809914000026707309915, 4.76034866808911937907595078316, 5.49541321575274865302432080349, 6.47373761562333669165555384503, 7.29887125480504178640242915193, 7.988458870531575125301197797382, 8.352440006252109968846501658767

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