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Properties

Label 2-1911-273.233-c0-0-3
Degree $2$
Conductor $1911$
Sign $-0.749 + 0.661i$
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  + (−0.923 + 1.60i)2-s + (0.5 + 0.866i)3-s + (−1.20 − 2.09i)4-s + (−0.382 + 0.662i)5-s − 1.84·6-s + 2.61·8-s + (−0.499 + 0.866i)9-s + (−0.707 − 1.22i)10-s + (0.923 + 1.60i)11-s + (1.20 − 2.09i)12-s + 13-s − 0.765·15-s + (−1.20 + 2.09i)16-s + (−0.923 − 1.60i)18-s + 1.84·20-s + ⋯
L(s)  = 1  + (−0.923 + 1.60i)2-s + (0.5 + 0.866i)3-s + (−1.20 − 2.09i)4-s + (−0.382 + 0.662i)5-s − 1.84·6-s + 2.61·8-s + (−0.499 + 0.866i)9-s + (−0.707 − 1.22i)10-s + (0.923 + 1.60i)11-s + (1.20 − 2.09i)12-s + 13-s − 0.765·15-s + (−1.20 + 2.09i)16-s + (−0.923 − 1.60i)18-s + 1.84·20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.749 + 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.749 + 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1911\)    =    \(3 \cdot 7^{2} \cdot 13\)
Sign: $-0.749 + 0.661i$
Analytic conductor: \(0.953713\)
Root analytic conductor: \(0.976582\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1911} (1598, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1911,\ (\ :0),\ -0.749 + 0.661i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7519954992\)
\(L(\frac12)\) \(\approx\) \(0.7519954992\)
\(L(1)\) 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 \( 1 - T \)
good2 \( 1 + (0.923 - 1.60i)T + (-0.5 - 0.866i)T^{2} \)
5 \( 1 + (0.382 - 0.662i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.923 - 1.60i)T + (-0.5 + 0.866i)T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 + 0.765T + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (-0.382 + 0.662i)T + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.923 + 1.60i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 - 0.765T + T^{2} \)
73 \( 1 + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + 1.84T + T^{2} \)
89 \( 1 + (-0.923 + 1.60i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 - T^{2} \)
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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

−9.567155583831095216227620655257, −9.028067678763218433189207940159, −8.311813921895301980927253514665, −7.53285517871671304935692086162, −6.91742850218327772254750961211, −6.20788870688983330763746574472, −5.15082448594077081253000767930, −4.38186146369477041218188486191, −3.46524360711148128865443834200, −1.76375855675621893460908717525, 0.814702987926618466448369048128, 1.46260479305835842940144171300, 2.77450387828792779595231365241, 3.53294508747077027329198344780, 4.19955118102855937218294304662, 5.80020887748496111647583058435, 6.76762976056683919550940094967, 7.977245133919768126180491054627, 8.403550367559921496207428346541, 8.902715912649067292311107849706

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