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Properties

Label 2-1911-1911.74-c0-0-0
Degree $2$
Conductor $1911$
Sign $0.489 + 0.871i$
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  + (−0.988 − 0.149i)3-s + (−0.900 − 0.433i)4-s + (0.955 − 0.294i)7-s + (0.955 + 0.294i)9-s + (0.826 + 0.563i)12-s + (0.0747 − 0.997i)13-s + (0.623 + 0.781i)16-s + (−0.955 + 1.65i)19-s + (−0.988 + 0.149i)21-s + (0.955 + 0.294i)25-s + (−0.900 − 0.433i)27-s + (−0.988 − 0.149i)28-s + (0.900 − 1.56i)31-s + (−0.733 − 0.680i)36-s + (−0.134 + 0.0648i)37-s + ⋯
L(s)  = 1  + (−0.988 − 0.149i)3-s + (−0.900 − 0.433i)4-s + (0.955 − 0.294i)7-s + (0.955 + 0.294i)9-s + (0.826 + 0.563i)12-s + (0.0747 − 0.997i)13-s + (0.623 + 0.781i)16-s + (−0.955 + 1.65i)19-s + (−0.988 + 0.149i)21-s + (0.955 + 0.294i)25-s + (−0.900 − 0.433i)27-s + (−0.988 − 0.149i)28-s + (0.900 − 1.56i)31-s + (−0.733 − 0.680i)36-s + (−0.134 + 0.0648i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.489 + 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.489 + 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1911\)    =    \(3 \cdot 7^{2} \cdot 13\)
Sign: $0.489 + 0.871i$
Analytic conductor: \(0.953713\)
Root analytic conductor: \(0.976582\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1911} (74, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1911,\ (\ :0),\ 0.489 + 0.871i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7246151395\)
\(L(\frac12)\) \(\approx\) \(0.7246151395\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.988 + 0.149i)T \)
7 \( 1 + (-0.955 + 0.294i)T \)
13 \( 1 + (-0.0747 + 0.997i)T \)
good2 \( 1 + (0.900 + 0.433i)T^{2} \)
5 \( 1 + (-0.955 - 0.294i)T^{2} \)
11 \( 1 + (-0.826 + 0.563i)T^{2} \)
17 \( 1 + (-0.623 + 0.781i)T^{2} \)
19 \( 1 + (0.955 - 1.65i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.623 - 0.781i)T^{2} \)
29 \( 1 + (0.988 + 0.149i)T^{2} \)
31 \( 1 + (-0.900 + 1.56i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.134 - 0.0648i)T + (0.623 - 0.781i)T^{2} \)
41 \( 1 + (0.733 - 0.680i)T^{2} \)
43 \( 1 + (-0.698 + 1.77i)T + (-0.733 - 0.680i)T^{2} \)
47 \( 1 + (-0.826 + 0.563i)T^{2} \)
53 \( 1 + (-0.365 + 0.930i)T^{2} \)
59 \( 1 + (0.222 - 0.974i)T^{2} \)
61 \( 1 + (-0.123 + 1.64i)T + (-0.988 - 0.149i)T^{2} \)
67 \( 1 + (0.623 + 1.07i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.988 - 0.149i)T^{2} \)
73 \( 1 + (-1.57 - 0.487i)T + (0.826 + 0.563i)T^{2} \)
79 \( 1 + (-0.900 - 1.56i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.900 - 0.433i)T^{2} \)
89 \( 1 + (0.900 - 0.433i)T^{2} \)
97 \( 1 + (0.955 + 1.65i)T + (-0.5 + 0.866i)T^{2} \)
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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

−9.409022103403168027593867165865, −8.199295251074525604349237914378, −7.976903289468201750241284373391, −6.76823145740771946659810332918, −5.81064279379799395448343575738, −5.34058420859418830514926866605, −4.47342819561844588717300562574, −3.78730068961762789270234792943, −1.92016009092220798625825229460, −0.77905750030103601339619921908, 1.17432496916898602143728876898, 2.70389083390457735616991958787, 4.18706446180606228951095700171, 4.66691867261709505394299034238, 5.21661281441907268328093208562, 6.40406032260064920623975983125, 7.04933178981734206119291144291, 8.074457667690092942352606224390, 8.889547206911793567570358118830, 9.308912632295539751655052634048

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