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Properties

Label 2-1407-1407.998-c0-0-0
Degree $2$
Conductor $1407$
Sign $0.959 + 0.283i$
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  + (0.723 − 0.690i)3-s + (0.415 + 0.909i)4-s + (−0.786 − 0.618i)7-s + (0.0475 − 0.998i)9-s + (0.928 + 0.371i)12-s + (1.56 + 0.625i)13-s + (−0.654 + 0.755i)16-s + (0.396 − 0.254i)19-s + (−0.995 + 0.0950i)21-s + (0.928 + 0.371i)25-s + (−0.654 − 0.755i)27-s + (0.235 − 0.971i)28-s + (0.186 − 1.29i)31-s + (0.928 − 0.371i)36-s − 1.57·37-s + ⋯
L(s)  = 1  + (0.723 − 0.690i)3-s + (0.415 + 0.909i)4-s + (−0.786 − 0.618i)7-s + (0.0475 − 0.998i)9-s + (0.928 + 0.371i)12-s + (1.56 + 0.625i)13-s + (−0.654 + 0.755i)16-s + (0.396 − 0.254i)19-s + (−0.995 + 0.0950i)21-s + (0.928 + 0.371i)25-s + (−0.654 − 0.755i)27-s + (0.235 − 0.971i)28-s + (0.186 − 1.29i)31-s + (0.928 − 0.371i)36-s − 1.57·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.959 + 0.283i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.959 + 0.283i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1407\)    =    \(3 \cdot 7 \cdot 67\)
Sign: $0.959 + 0.283i$
Analytic conductor: \(0.702184\)
Root analytic conductor: \(0.837964\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1407} (998, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1407,\ (\ :0),\ 0.959 + 0.283i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.470340036\)
\(L(\frac12)\) \(\approx\) \(1.470340036\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.723 + 0.690i)T \)
7 \( 1 + (0.786 + 0.618i)T \)
67 \( 1 + (0.786 - 0.618i)T \)
good2 \( 1 + (-0.415 - 0.909i)T^{2} \)
5 \( 1 + (-0.928 - 0.371i)T^{2} \)
11 \( 1 + (0.142 + 0.989i)T^{2} \)
13 \( 1 + (-1.56 - 0.625i)T + (0.723 + 0.690i)T^{2} \)
17 \( 1 + (-0.981 + 0.189i)T^{2} \)
19 \( 1 + (-0.396 + 0.254i)T + (0.415 - 0.909i)T^{2} \)
23 \( 1 + (-0.841 - 0.540i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (-0.186 + 1.29i)T + (-0.959 - 0.281i)T^{2} \)
37 \( 1 + 1.57T + T^{2} \)
41 \( 1 + (-0.981 + 0.189i)T^{2} \)
43 \( 1 + (0.118 - 0.258i)T + (-0.654 - 0.755i)T^{2} \)
47 \( 1 + (-0.0475 + 0.998i)T^{2} \)
53 \( 1 + (-0.981 - 0.189i)T^{2} \)
59 \( 1 + (-0.235 - 0.971i)T^{2} \)
61 \( 1 + (0.308 + 0.356i)T + (-0.142 + 0.989i)T^{2} \)
71 \( 1 + (0.327 + 0.945i)T^{2} \)
73 \( 1 + (0.642 - 1.85i)T + (-0.786 - 0.618i)T^{2} \)
79 \( 1 + (1.45 + 0.584i)T + (0.723 + 0.690i)T^{2} \)
83 \( 1 + (0.786 - 0.618i)T^{2} \)
89 \( 1 + (-0.0475 - 0.998i)T^{2} \)
97 \( 1 + (0.723 - 1.25i)T + (-0.5 - 0.866i)T^{2} \)
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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

−9.434178332082661928631994514852, −8.783056351249595292996107804507, −8.120302925825081690631685003702, −7.17821838808293972128318150671, −6.76840833498785337017485904096, −5.95174419969584695148872774657, −4.18562216923995715454691823730, −3.52454819097518152112372555188, −2.76233827813460515813571684459, −1.42986227353894751810014168920, 1.56181742358937458308982330241, 2.88175185648265899783932716200, 3.50968258410974300799801610622, 4.84849367212256253117415082064, 5.65030485715970432359525427710, 6.38131807894816025540260054823, 7.32872177254567904962282173280, 8.657920331221584426653593327366, 8.824069701061931715559752884752, 9.904076965156001015053987352768

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