Properties

Label 2-1407-1407.1286-c0-0-0
Degree $2$
Conductor $1407$
Sign $0.966 - 0.256i$
Analytic cond. $0.702184$
Root an. cond. $0.837964$
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  + (0.981 + 0.189i)3-s + (0.959 − 0.281i)4-s + (−0.580 + 0.814i)7-s + (0.928 + 0.371i)9-s + (0.995 − 0.0950i)12-s + (0.283 − 0.0270i)13-s + (0.841 − 0.540i)16-s + (−1.87 − 0.268i)19-s + (−0.723 + 0.690i)21-s + (−0.995 + 0.0950i)25-s + (0.841 + 0.540i)27-s + (−0.327 + 0.945i)28-s + (−0.698 − 1.53i)31-s + (0.995 + 0.0950i)36-s + 1.16·37-s + ⋯
L(s)  = 1  + (0.981 + 0.189i)3-s + (0.959 − 0.281i)4-s + (−0.580 + 0.814i)7-s + (0.928 + 0.371i)9-s + (0.995 − 0.0950i)12-s + (0.283 − 0.0270i)13-s + (0.841 − 0.540i)16-s + (−1.87 − 0.268i)19-s + (−0.723 + 0.690i)21-s + (−0.995 + 0.0950i)25-s + (0.841 + 0.540i)27-s + (−0.327 + 0.945i)28-s + (−0.698 − 1.53i)31-s + (0.995 + 0.0950i)36-s + 1.16·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 - 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 - 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1407\)    =    \(3 \cdot 7 \cdot 67\)
Sign: $0.966 - 0.256i$
Analytic conductor: \(0.702184\)
Root analytic conductor: \(0.837964\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1407} (1286, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1407,\ (\ :0),\ 0.966 - 0.256i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.740788288\)
\(L(\frac12)\) \(\approx\) \(1.740788288\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.981 - 0.189i)T \)
7 \( 1 + (0.580 - 0.814i)T \)
67 \( 1 + (0.580 + 0.814i)T \)
good2 \( 1 + (-0.959 + 0.281i)T^{2} \)
5 \( 1 + (0.995 - 0.0950i)T^{2} \)
11 \( 1 + (0.415 - 0.909i)T^{2} \)
13 \( 1 + (-0.283 + 0.0270i)T + (0.981 - 0.189i)T^{2} \)
17 \( 1 + (0.0475 - 0.998i)T^{2} \)
19 \( 1 + (1.87 + 0.268i)T + (0.959 + 0.281i)T^{2} \)
23 \( 1 + (0.142 + 0.989i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.698 + 1.53i)T + (-0.654 + 0.755i)T^{2} \)
37 \( 1 - 1.16T + T^{2} \)
41 \( 1 + (-0.0475 + 0.998i)T^{2} \)
43 \( 1 + (0.512 - 1.74i)T + (-0.841 - 0.540i)T^{2} \)
47 \( 1 + (0.928 + 0.371i)T^{2} \)
53 \( 1 + (0.0475 + 0.998i)T^{2} \)
59 \( 1 + (-0.327 - 0.945i)T^{2} \)
61 \( 1 + (-0.550 - 0.353i)T + (0.415 + 0.909i)T^{2} \)
71 \( 1 + (0.888 + 0.458i)T^{2} \)
73 \( 1 + (0.915 + 1.77i)T + (-0.580 + 0.814i)T^{2} \)
79 \( 1 + (0.154 + 1.62i)T + (-0.981 + 0.189i)T^{2} \)
83 \( 1 + (0.580 + 0.814i)T^{2} \)
89 \( 1 + (0.928 - 0.371i)T^{2} \)
97 \( 1 + (-0.981 - 1.70i)T + (-0.5 + 0.866i)T^{2} \)
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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

−9.679239852315826427677054244793, −9.073052185129314283952796735921, −8.147752033314469301319883763247, −7.50659883240910368521310768294, −6.37283243864299123029433400751, −5.98741344586093731762677011012, −4.59601184212343384923929890670, −3.54174048821990545433198042873, −2.55466128502101915617744735658, −1.90759502706131066113973381323, 1.61995979299475932804378228882, 2.59764643311395038830793457544, 3.62223116961172944337116143812, 4.19971468982550385574794322258, 5.86944770628810252383501161654, 6.78359327232243857438513941602, 7.16878729697214249886953182944, 8.157770471474146066149830907125, 8.687005828737489226596843480091, 9.822164947040527159020669739213

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