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Properties

Label 2-1067-1067.626-c0-0-0
Degree $2$
Conductor $1067$
Sign $-0.831 + 0.555i$
Analytic cond. $0.532502$
Root an. cond. $0.729727$
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  + (−0.315 − 0.410i)3-s + (−0.965 − 0.258i)4-s + (0.641 − 1.88i)5-s + (0.189 − 0.707i)9-s + (−0.991 + 0.130i)11-s + (0.198 + 0.478i)12-s + (−0.978 + 0.332i)15-s + (0.866 + 0.499i)16-s + (−1.10 + 1.65i)20-s + (−0.123 + 1.88i)23-s + (−2.36 − 1.81i)25-s + (−0.828 + 0.343i)27-s + (1.46 − 1.12i)31-s + (0.366 + 0.366i)33-s + (−0.366 + 0.633i)36-s + (−1.50 + 0.0983i)37-s + ⋯
L(s)  = 1  + (−0.315 − 0.410i)3-s + (−0.965 − 0.258i)4-s + (0.641 − 1.88i)5-s + (0.189 − 0.707i)9-s + (−0.991 + 0.130i)11-s + (0.198 + 0.478i)12-s + (−0.978 + 0.332i)15-s + (0.866 + 0.499i)16-s + (−1.10 + 1.65i)20-s + (−0.123 + 1.88i)23-s + (−2.36 − 1.81i)25-s + (−0.828 + 0.343i)27-s + (1.46 − 1.12i)31-s + (0.366 + 0.366i)33-s + (−0.366 + 0.633i)36-s + (−1.50 + 0.0983i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1067 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.831 + 0.555i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1067 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.831 + 0.555i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1067\)    =    \(11 \cdot 97\)
Sign: $-0.831 + 0.555i$
Analytic conductor: \(0.532502\)
Root analytic conductor: \(0.729727\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1067} (626, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1067,\ (\ :0),\ -0.831 + 0.555i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6980168376\)
\(L(\frac12)\) \(\approx\) \(0.6980168376\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (0.991 - 0.130i)T \)
97 \( 1 + (-0.130 + 0.991i)T \)
good2 \( 1 + (0.965 + 0.258i)T^{2} \)
3 \( 1 + (0.315 + 0.410i)T + (-0.258 + 0.965i)T^{2} \)
5 \( 1 + (-0.641 + 1.88i)T + (-0.793 - 0.608i)T^{2} \)
7 \( 1 + (0.130 + 0.991i)T^{2} \)
13 \( 1 + (-0.793 - 0.608i)T^{2} \)
17 \( 1 + (-0.130 + 0.991i)T^{2} \)
19 \( 1 + (-0.923 + 0.382i)T^{2} \)
23 \( 1 + (0.123 - 1.88i)T + (-0.991 - 0.130i)T^{2} \)
29 \( 1 + (-0.608 + 0.793i)T^{2} \)
31 \( 1 + (-1.46 + 1.12i)T + (0.258 - 0.965i)T^{2} \)
37 \( 1 + (1.50 - 0.0983i)T + (0.991 - 0.130i)T^{2} \)
41 \( 1 + (0.608 - 0.793i)T^{2} \)
43 \( 1 + (0.866 - 0.5i)T^{2} \)
47 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
53 \( 1 + (-1.57 - 0.207i)T + (0.965 + 0.258i)T^{2} \)
59 \( 1 + (0.00855 + 0.130i)T + (-0.991 + 0.130i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.172 + 0.867i)T + (-0.923 + 0.382i)T^{2} \)
71 \( 1 + (0.583 - 1.18i)T + (-0.608 - 0.793i)T^{2} \)
73 \( 1 + (-0.866 + 0.5i)T^{2} \)
79 \( 1 + (-0.707 - 0.707i)T^{2} \)
83 \( 1 + (0.130 - 0.991i)T^{2} \)
89 \( 1 + (-0.707 - 0.707i)T^{2} \)
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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

−9.782242653081118139084823393569, −8.908494253689045268957713909774, −8.370034116144784166184864543169, −7.40707200712702541457459941957, −5.93062536699697920817000074705, −5.47747356238118997232237771954, −4.73332733853850570149472805215, −3.77872346160479207066231136152, −1.79911545265745522051768294292, −0.69805277024854729059933214646, 2.38034685875807529743675520512, 3.16218551199615786345919636631, 4.37481529785644290273954845546, 5.26808750236861673831883520120, 6.14467837697166457813463153359, 7.09794082773466683752650685268, 7.907309718239333945884292135104, 8.817947124072794807869639579480, 10.00189860798301851386677803540, 10.44794224385588224124759811082

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