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Properties

Label 2-1407-1407.647-c0-0-0
Degree $2$
Conductor $1407$
Sign $-0.233 - 0.972i$
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  + (0.786 + 0.618i)3-s + (−0.0475 + 0.998i)4-s + (−0.959 + 0.281i)7-s + (0.235 + 0.971i)9-s + (−0.654 + 0.755i)12-s + (1.88 + 0.363i)13-s + (−0.995 − 0.0950i)16-s + (−1.20 − 0.291i)19-s + (−0.928 − 0.371i)21-s + (−0.327 − 0.945i)25-s + (−0.415 + 0.909i)27-s + (−0.235 − 0.971i)28-s + (−0.271 + 0.785i)31-s + (−0.981 + 0.189i)36-s + (0.327 − 0.566i)37-s + ⋯
L(s)  = 1  + (0.786 + 0.618i)3-s + (−0.0475 + 0.998i)4-s + (−0.959 + 0.281i)7-s + (0.235 + 0.971i)9-s + (−0.654 + 0.755i)12-s + (1.88 + 0.363i)13-s + (−0.995 − 0.0950i)16-s + (−1.20 − 0.291i)19-s + (−0.928 − 0.371i)21-s + (−0.327 − 0.945i)25-s + (−0.415 + 0.909i)27-s + (−0.235 − 0.971i)28-s + (−0.271 + 0.785i)31-s + (−0.981 + 0.189i)36-s + (0.327 − 0.566i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.233 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.233 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1407\)    =    \(3 \cdot 7 \cdot 67\)
Sign: $-0.233 - 0.972i$
Analytic conductor: \(0.702184\)
Root analytic conductor: \(0.837964\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1407} (647, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1407,\ (\ :0),\ -0.233 - 0.972i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.270541953\)
\(L(\frac12)\) \(\approx\) \(1.270541953\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.786 - 0.618i)T \)
7 \( 1 + (0.959 - 0.281i)T \)
67 \( 1 + (-0.654 + 0.755i)T \)
good2 \( 1 + (0.0475 - 0.998i)T^{2} \)
5 \( 1 + (0.327 + 0.945i)T^{2} \)
11 \( 1 + (-0.327 - 0.945i)T^{2} \)
13 \( 1 + (-1.88 - 0.363i)T + (0.928 + 0.371i)T^{2} \)
17 \( 1 + (0.415 - 0.909i)T^{2} \)
19 \( 1 + (1.20 + 0.291i)T + (0.888 + 0.458i)T^{2} \)
23 \( 1 + (-0.235 - 0.971i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.271 - 0.785i)T + (-0.786 - 0.618i)T^{2} \)
37 \( 1 + (-0.327 + 0.566i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.995 - 0.0950i)T^{2} \)
43 \( 1 + (-0.817 - 1.27i)T + (-0.415 + 0.909i)T^{2} \)
47 \( 1 + (-0.959 - 0.281i)T^{2} \)
53 \( 1 + (0.580 - 0.814i)T^{2} \)
59 \( 1 + (0.928 - 0.371i)T^{2} \)
61 \( 1 + (0.911 + 1.28i)T + (-0.327 + 0.945i)T^{2} \)
71 \( 1 + (-0.580 + 0.814i)T^{2} \)
73 \( 1 + (-0.172 - 0.0789i)T + (0.654 + 0.755i)T^{2} \)
79 \( 1 + (-1.42 - 1.23i)T + (0.142 + 0.989i)T^{2} \)
83 \( 1 + (-0.327 - 0.945i)T^{2} \)
89 \( 1 + (0.235 - 0.971i)T^{2} \)
97 \( 1 + (0.928 + 1.60i)T + (-0.5 + 0.866i)T^{2} \)
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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

−9.765295094926391480916474934347, −8.971021773168483357336808545441, −8.569308753670304808633258908262, −7.83771561487837779888260933210, −6.71751105997192568982168045061, −6.04713331776416825658455575046, −4.56274625560381684096056589321, −3.83003002379179720804677038036, −3.18154590878898206174664022516, −2.16838607416288083456682158732, 1.01696196587351401433076820678, 2.16723592991673922730858451081, 3.44519642573432883094806340792, 4.14720317745348322960639233306, 5.76318832092751535585329097423, 6.21852534400727387031809440806, 6.96039507743557364248677434054, 7.976416302491738377736547946129, 8.888667341240690966745075953114, 9.323267719741722307759937937066

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