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Properties

Label 2-1407-1407.353-c0-0-0
Degree $2$
Conductor $1407$
Sign $0.115 + 0.993i$
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  + (0.888 − 0.458i)3-s + (−0.981 − 0.189i)4-s + (0.415 − 0.909i)7-s + (0.580 − 0.814i)9-s + (−0.959 + 0.281i)12-s + (−0.601 − 0.573i)13-s + (0.928 + 0.371i)16-s + (−0.746 + 0.531i)19-s + (−0.0475 − 0.998i)21-s + (0.235 − 0.971i)25-s + (0.142 − 0.989i)27-s + (−0.580 + 0.814i)28-s + (−0.0671 − 0.276i)31-s + (−0.723 + 0.690i)36-s + (−0.235 + 0.408i)37-s + ⋯
L(s)  = 1  + (0.888 − 0.458i)3-s + (−0.981 − 0.189i)4-s + (0.415 − 0.909i)7-s + (0.580 − 0.814i)9-s + (−0.959 + 0.281i)12-s + (−0.601 − 0.573i)13-s + (0.928 + 0.371i)16-s + (−0.746 + 0.531i)19-s + (−0.0475 − 0.998i)21-s + (0.235 − 0.971i)25-s + (0.142 − 0.989i)27-s + (−0.580 + 0.814i)28-s + (−0.0671 − 0.276i)31-s + (−0.723 + 0.690i)36-s + (−0.235 + 0.408i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.115 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.115 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1407\)    =    \(3 \cdot 7 \cdot 67\)
Sign: $0.115 + 0.993i$
Analytic conductor: \(0.702184\)
Root analytic conductor: \(0.837964\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1407} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1407,\ (\ :0),\ 0.115 + 0.993i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.162625953\)
\(L(\frac12)\) \(\approx\) \(1.162625953\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.888 + 0.458i)T \)
7 \( 1 + (-0.415 + 0.909i)T \)
67 \( 1 + (-0.959 + 0.281i)T \)
good2 \( 1 + (0.981 + 0.189i)T^{2} \)
5 \( 1 + (-0.235 + 0.971i)T^{2} \)
11 \( 1 + (0.235 - 0.971i)T^{2} \)
13 \( 1 + (0.601 + 0.573i)T + (0.0475 + 0.998i)T^{2} \)
17 \( 1 + (-0.142 + 0.989i)T^{2} \)
19 \( 1 + (0.746 - 0.531i)T + (0.327 - 0.945i)T^{2} \)
23 \( 1 + (-0.580 + 0.814i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.0671 + 0.276i)T + (-0.888 + 0.458i)T^{2} \)
37 \( 1 + (0.235 - 0.408i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.928 + 0.371i)T^{2} \)
43 \( 1 + (-0.425 + 0.368i)T + (0.142 - 0.989i)T^{2} \)
47 \( 1 + (0.415 + 0.909i)T^{2} \)
53 \( 1 + (-0.786 + 0.618i)T^{2} \)
59 \( 1 + (0.0475 - 0.998i)T^{2} \)
61 \( 1 + (-1.39 - 1.09i)T + (0.235 + 0.971i)T^{2} \)
71 \( 1 + (0.786 - 0.618i)T^{2} \)
73 \( 1 + (-0.735 - 0.105i)T + (0.959 + 0.281i)T^{2} \)
79 \( 1 + (-0.547 - 1.86i)T + (-0.841 + 0.540i)T^{2} \)
83 \( 1 + (0.235 - 0.971i)T^{2} \)
89 \( 1 + (0.580 + 0.814i)T^{2} \)
97 \( 1 + (0.0475 + 0.0824i)T + (-0.5 + 0.866i)T^{2} \)
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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

−9.617904976192744793013580719464, −8.519193208856142774185008955122, −8.163577102984499323476011626391, −7.34058013838882674778409715668, −6.43122526882868872907212172562, −5.24723479022078069685677174277, −4.28674049234432471056129354577, −3.65601521723037674037976026898, −2.33647211547592265910259641821, −0.950864833001833931104554822445, 1.92562660989603884291954105874, 2.97459113822319959695608531071, 4.01538145577704474133803062509, 4.81760378023826406374635836728, 5.44666770557872379753536791640, 6.84854656150849662897574744011, 7.85140772621068208116305153795, 8.482274508990263785960468102374, 9.203703715466807448402215690351, 9.511467040668046172034859817477

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