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Properties

Label 2-1407-1407.116-c0-0-0
Degree $2$
Conductor $1407$
Sign $0.999 + 0.00622i$
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 + (−1.03 − 1.44i)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 + (−1.03 − 1.44i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00622i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00622i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1407\)    =    \(3 \cdot 7 \cdot 67\)
Sign: $0.999 + 0.00622i$
Analytic conductor: \(0.702184\)
Root analytic conductor: \(0.837964\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1407} (116, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1407,\ (\ :0),\ 0.999 + 0.00622i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.106975021\)
\(L(\frac12)\) \(\approx\) \(1.106975021\)
\(L(1)\) 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 \( 1 + (1.03 + 1.44i)T + (-0.327 + 0.945i)T^{2} \)
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 \( 1 + (-1.25 - 1.45i)T + (-0.142 + 0.989i)T^{2} \)
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 \( 1 + (0.264 - 1.83i)T + (-0.959 - 0.281i)T^{2} \)
79 \( 1 + (0.452 - 0.132i)T + (0.841 - 0.540i)T^{2} \)
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

−10.07488098590250541679003054715, −8.987875555738077227507700346557, −8.053311510752162561150180772870, −6.97750872042254562110571262422, −6.64146248321226310750212304194, −5.73775641852957166136732546475, −4.54285290839912422321502919718, −3.99658216325276118461378987824, −2.62925145468953252965656926339, −1.20099174954631601048458790066, 1.47929762579004499911982925693, 2.30098059330526078840131179164, 3.66461522392361870463652220119, 5.09595693132725506974963557356, 5.81127286444563675266329956228, 6.26063939756579625155331219956, 7.27347739297084454004147722739, 7.971869843473107868846921018508, 8.812083850537143392617203610005, 10.05194207212809251720007234247

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