Properties

Label 2-1911-1.1-c1-0-63
Degree $2$
Conductor $1911$
Sign $-1$
Analytic cond. $15.2594$
Root an. cond. $3.90632$
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  + 0.414·2-s − 3-s − 1.82·4-s + 2.82·5-s − 0.414·6-s − 1.58·8-s + 9-s + 1.17·10-s − 2·11-s + 1.82·12-s + 13-s − 2.82·15-s + 3·16-s − 7.65·17-s + 0.414·18-s + 2.82·19-s − 5.17·20-s − 0.828·22-s − 4·23-s + 1.58·24-s + 3.00·25-s + 0.414·26-s − 27-s + 2·29-s − 1.17·30-s + 1.17·31-s + 4.41·32-s + ⋯
L(s)  = 1  + 0.292·2-s − 0.577·3-s − 0.914·4-s + 1.26·5-s − 0.169·6-s − 0.560·8-s + 0.333·9-s + 0.370·10-s − 0.603·11-s + 0.527·12-s + 0.277·13-s − 0.730·15-s + 0.750·16-s − 1.85·17-s + 0.0976·18-s + 0.648·19-s − 1.15·20-s − 0.176·22-s − 0.834·23-s + 0.323·24-s + 0.600·25-s + 0.0812·26-s − 0.192·27-s + 0.371·29-s − 0.213·30-s + 0.210·31-s + 0.780·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1911\)    =    \(3 \cdot 7^{2} \cdot 13\)
Sign: $-1$
Analytic conductor: \(15.2594\)
Root analytic conductor: \(3.90632\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1911} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1911,\ (\ :1/2),\ -1)\)

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)$
bad3 \( 1 + T \)
7 \( 1 \)
13 \( 1 - T \)
good2 \( 1 - 0.414T + 2T^{2} \)
5 \( 1 - 2.82T + 5T^{2} \)
11 \( 1 + 2T + 11T^{2} \)
17 \( 1 + 7.65T + 17T^{2} \)
19 \( 1 - 2.82T + 19T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 - 1.17T + 31T^{2} \)
37 \( 1 + 7.65T + 37T^{2} \)
41 \( 1 + 5.17T + 41T^{2} \)
43 \( 1 + 1.65T + 43T^{2} \)
47 \( 1 - 11.6T + 47T^{2} \)
53 \( 1 + 2T + 53T^{2} \)
59 \( 1 + 7.65T + 59T^{2} \)
61 \( 1 + 13.3T + 61T^{2} \)
67 \( 1 - 6.82T + 67T^{2} \)
71 \( 1 - 2T + 71T^{2} \)
73 \( 1 + 0.343T + 73T^{2} \)
79 \( 1 + 11.3T + 79T^{2} \)
83 \( 1 + 3.65T + 83T^{2} \)
89 \( 1 + 14.8T + 89T^{2} \)
97 \( 1 + 3.65T + 97T^{2} \)
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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

−8.958223884801740312196084734694, −8.212382684805168410662833884574, −7.00971164145282174462860886388, −6.17229203052613060875050123090, −5.56784258456527622396747878423, −4.86777315963195236130899693653, −4.06088834140592110241002946061, −2.76439236338286257351377342696, −1.61542267836316041388675843124, 0, 1.61542267836316041388675843124, 2.76439236338286257351377342696, 4.06088834140592110241002946061, 4.86777315963195236130899693653, 5.56784258456527622396747878423, 6.17229203052613060875050123090, 7.00971164145282174462860886388, 8.212382684805168410662833884574, 8.958223884801740312196084734694

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