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Properties

Label 2-1197-7.2-c1-0-49
Degree $2$
Conductor $1197$
Sign $0.386 + 0.922i$
Analytic cond. $9.55809$
Root an. cond. $3.09161$
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 + 1.73i)4-s + (1 − 1.73i)5-s + (0.5 − 2.59i)7-s + (−1.5 − 2.59i)11-s + 2·13-s + (−1.99 + 3.46i)16-s + (−3.5 − 6.06i)17-s + (0.5 − 0.866i)19-s + 3.99·20-s + (2.5 − 4.33i)23-s + (0.500 + 0.866i)25-s + (5 − 1.73i)28-s − 2·29-s + (−5 − 8.66i)31-s + (−4 − 3.46i)35-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)4-s + (0.447 − 0.774i)5-s + (0.188 − 0.981i)7-s + (−0.452 − 0.783i)11-s + 0.554·13-s + (−0.499 + 0.866i)16-s + (−0.848 − 1.47i)17-s + (0.114 − 0.198i)19-s + 0.894·20-s + (0.521 − 0.902i)23-s + (0.100 + 0.173i)25-s + (0.944 − 0.327i)28-s − 0.371·29-s + (−0.898 − 1.55i)31-s + (−0.676 − 0.585i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1197 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1197 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1197\)    =    \(3^{2} \cdot 7 \cdot 19\)
Sign: $0.386 + 0.922i$
Analytic conductor: \(9.55809\)
Root analytic conductor: \(3.09161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1197} (856, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1197,\ (\ :1/2),\ 0.386 + 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.821548284\)
\(L(\frac12)\) \(\approx\) \(1.821548284\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (-0.5 + 2.59i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (-1 - 1.73i)T^{2} \)
5 \( 1 + (-1 + 1.73i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.5 + 2.59i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + (3.5 + 6.06i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-2.5 + 4.33i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 + (5 + 8.66i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4 - 6.92i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 - 12T + 43T^{2} \)
47 \( 1 + (2.5 - 4.33i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2 - 3.46i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-7 - 12.1i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.5 + 11.2i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1 - 1.73i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 10T + 71T^{2} \)
73 \( 1 + (0.5 + 0.866i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2 + 3.46i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 9T + 83T^{2} \)
89 \( 1 + (-9 + 15.5i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 6T + 97T^{2} \)
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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

−9.380402981580854172671331157514, −8.784198014147347797834613386619, −7.939792917997030024454190316784, −7.19834983444193913271334574946, −6.40269430681634937224962408065, −5.25021013409811545626589968394, −4.37708533545215457938505764914, −3.36949448772577536712305346600, −2.28271469726049107960112444562, −0.75970907950683873875276380234, 1.75372935083218540618336424563, 2.32661254643518417828854216703, 3.63407390337203064714297090050, 5.12836215565507625946555648022, 5.69253610454756017364130226273, 6.55986678383901380308617186732, 7.15794408352459252249858922352, 8.387954133710412312981801478666, 9.186089208171209927997694413526, 10.01475256370326815411411489735

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