Properties

Label 2-1911-273.74-c0-0-1
Degree $2$
Conductor $1911$
Sign $0.617 - 0.786i$
Analytic cond. $0.953713$
Root an. cond. $0.976582$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 0.866i)3-s + 4-s + (−0.499 + 0.866i)9-s + (0.5 + 0.866i)12-s + (0.5 − 0.866i)13-s + 16-s + (−0.5 + 0.866i)19-s + (−0.5 + 0.866i)25-s − 0.999·27-s + (1 − 1.73i)31-s + (−0.499 + 0.866i)36-s − 37-s + 0.999·39-s + (0.5 + 0.866i)43-s + (0.5 + 0.866i)48-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)3-s + 4-s + (−0.499 + 0.866i)9-s + (0.5 + 0.866i)12-s + (0.5 − 0.866i)13-s + 16-s + (−0.5 + 0.866i)19-s + (−0.5 + 0.866i)25-s − 0.999·27-s + (1 − 1.73i)31-s + (−0.499 + 0.866i)36-s − 37-s + 0.999·39-s + (0.5 + 0.866i)43-s + (0.5 + 0.866i)48-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.617 - 0.786i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.617 - 0.786i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1911\)    =    \(3 \cdot 7^{2} \cdot 13\)
Sign: $0.617 - 0.786i$
Analytic conductor: \(0.953713\)
Root analytic conductor: \(0.976582\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1911} (1439, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1911,\ (\ :0),\ 0.617 - 0.786i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.696296276\)
\(L(\frac12)\) \(\approx\) \(1.696296276\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 \)
13 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 - T^{2} \)
5 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + T + T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T^{2} \)
73 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
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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

−9.683658150700785357083403861823, −8.658113727467504118705857677477, −7.954417759635348774516630012654, −7.38917850467272199772065136890, −6.08638072854603139718020848497, −5.70125092543156735846796610153, −4.47847902078195907124749428176, −3.54268037307200548863143341021, −2.81365633739104343889341931335, −1.73041671130129816161272089754, 1.35722756020361476129655897864, 2.28814900804269239706097991249, 3.10631121553068922496668328592, 4.18042676357661467957507815636, 5.49893730684909701658459104973, 6.57904853194271840348577926976, 6.71801570451689836103093637457, 7.63236998634976486279449406144, 8.492599284537430327152719892486, 8.996826493859654810548368651477

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