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Properties

Label 2-1911-13.12-c1-0-0
Degree $2$
Conductor $1911$
Sign $0.410 + 0.911i$
Analytic cond. $15.2594$
Root an. cond. $3.90632$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.48i·2-s − 3-s − 0.193·4-s + 4.15i·5-s − 1.48i·6-s + 2.67i·8-s + 9-s − 6.15·10-s − 3.19i·11-s + 0.193·12-s + (−1.48 − 3.28i)13-s − 4.15i·15-s − 4.35·16-s − 3.35·17-s + 1.48i·18-s − 2.38i·19-s + ⋯
L(s)  = 1  + 1.04i·2-s − 0.577·3-s − 0.0969·4-s + 1.85i·5-s − 0.604i·6-s + 0.945i·8-s + 0.333·9-s − 1.94·10-s − 0.963i·11-s + 0.0559·12-s + (−0.410 − 0.911i)13-s − 1.07i·15-s − 1.08·16-s − 0.812·17-s + 0.349i·18-s − 0.547i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.410 + 0.911i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.410 + 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1911\)    =    \(3 \cdot 7^{2} \cdot 13\)
Sign: $0.410 + 0.911i$
Analytic conductor: \(15.2594\)
Root analytic conductor: \(3.90632\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1911} (883, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1911,\ (\ :1/2),\ 0.410 + 0.911i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1383194752\)
\(L(\frac12)\) \(\approx\) \(0.1383194752\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.48 + 3.28i)T \)
good2 \( 1 - 1.48iT - 2T^{2} \)
5 \( 1 - 4.15iT - 5T^{2} \)
11 \( 1 + 3.19iT - 11T^{2} \)
17 \( 1 + 3.35T + 17T^{2} \)
19 \( 1 + 2.38iT - 19T^{2} \)
23 \( 1 - 0.387T + 23T^{2} \)
29 \( 1 + 7.92T + 29T^{2} \)
31 \( 1 + 10.7iT - 31T^{2} \)
37 \( 1 + 1.61iT - 37T^{2} \)
41 \( 1 + 1.45iT - 41T^{2} \)
43 \( 1 - 1.92T + 43T^{2} \)
47 \( 1 - 3.76iT - 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 - 6.15iT - 59T^{2} \)
61 \( 1 + 14.4T + 61T^{2} \)
67 \( 1 - 5.61iT - 67T^{2} \)
71 \( 1 - 11.8iT - 71T^{2} \)
73 \( 1 - 15.6iT - 73T^{2} \)
79 \( 1 + 8.96T + 79T^{2} \)
83 \( 1 + 6.99iT - 83T^{2} \)
89 \( 1 + 0.932iT - 89T^{2} \)
97 \( 1 - 3.35iT - 97T^{2} \)
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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

−9.941918735166031664149252935758, −8.924906288237989465269232271030, −7.76390828717565526411808748743, −7.42373397788222276127540892779, −6.66678857614951176514905661678, −5.94880250836217116333678227625, −5.59903555944447002956213963203, −4.21289177338175511976256952723, −3.03213426754001186191992816323, −2.29290838490209376026516058917, 0.05006247155408423926986687019, 1.52017363996875493607568444759, 1.91661352727798043243506738908, 3.59481036421459030761546156561, 4.61065575765795872003783602654, 4.84735414181332234720780930689, 6.07359402085860734872736703763, 6.98219337473295574473882349987, 7.83479066128507489051001126712, 9.087624368439693920090233367051

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