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Properties

Label 2-1407-1407.1025-c0-0-0
Degree $2$
Conductor $1407$
Sign $0.954 - 0.297i$
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 + (−0.841 − 0.540i)4-s + (−0.981 + 0.189i)7-s + (0.235 + 0.971i)9-s + (0.327 + 0.945i)12-s + (0.627 + 1.81i)13-s + (0.415 + 0.909i)16-s + (−0.209 − 0.713i)19-s + (0.888 + 0.458i)21-s + (−0.327 − 0.945i)25-s + (0.415 − 0.909i)27-s + (0.928 + 0.371i)28-s + (0.544 + 0.627i)31-s + (0.327 − 0.945i)36-s + 1.96·37-s + ⋯
L(s)  = 1  + (−0.786 − 0.618i)3-s + (−0.841 − 0.540i)4-s + (−0.981 + 0.189i)7-s + (0.235 + 0.971i)9-s + (0.327 + 0.945i)12-s + (0.627 + 1.81i)13-s + (0.415 + 0.909i)16-s + (−0.209 − 0.713i)19-s + (0.888 + 0.458i)21-s + (−0.327 − 0.945i)25-s + (0.415 − 0.909i)27-s + (0.928 + 0.371i)28-s + (0.544 + 0.627i)31-s + (0.327 − 0.945i)36-s + 1.96·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.954 - 0.297i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.954 - 0.297i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1407\)    =    \(3 \cdot 7 \cdot 67\)
Sign: $0.954 - 0.297i$
Analytic conductor: \(0.702184\)
Root analytic conductor: \(0.837964\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1407} (1025, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1407,\ (\ :0),\ 0.954 - 0.297i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5394843065\)
\(L(\frac12)\) \(\approx\) \(0.5394843065\)
\(L(1)\) 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 \( 1 + (0.981 - 0.189i)T \)
67 \( 1 + (0.981 + 0.189i)T \)
good2 \( 1 + (0.841 + 0.540i)T^{2} \)
5 \( 1 + (0.327 + 0.945i)T^{2} \)
11 \( 1 + (-0.654 + 0.755i)T^{2} \)
13 \( 1 + (-0.627 - 1.81i)T + (-0.786 + 0.618i)T^{2} \)
17 \( 1 + (0.580 + 0.814i)T^{2} \)
19 \( 1 + (0.209 + 0.713i)T + (-0.841 + 0.540i)T^{2} \)
23 \( 1 + (0.959 + 0.281i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (-0.544 - 0.627i)T + (-0.142 + 0.989i)T^{2} \)
37 \( 1 - 1.96T + T^{2} \)
41 \( 1 + (-0.580 - 0.814i)T^{2} \)
43 \( 1 + (-0.817 - 1.27i)T + (-0.415 + 0.909i)T^{2} \)
47 \( 1 + (0.235 + 0.971i)T^{2} \)
53 \( 1 + (0.580 - 0.814i)T^{2} \)
59 \( 1 + (0.928 - 0.371i)T^{2} \)
61 \( 1 + (0.771 - 1.68i)T + (-0.654 - 0.755i)T^{2} \)
71 \( 1 + (0.995 + 0.0950i)T^{2} \)
73 \( 1 + (-0.154 - 1.62i)T + (-0.981 + 0.189i)T^{2} \)
79 \( 1 + (-0.357 + 0.123i)T + (0.786 - 0.618i)T^{2} \)
83 \( 1 + (0.981 + 0.189i)T^{2} \)
89 \( 1 + (0.235 - 0.971i)T^{2} \)
97 \( 1 + (0.786 - 1.36i)T + (-0.5 - 0.866i)T^{2} \)
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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

−9.700750198381854920024223769582, −9.122243733326807434219495793451, −8.291116724575876832325838340600, −7.11718058543501518921126304710, −6.27432360936979230219687036036, −5.98918315549168453831277301586, −4.68107181174604205664985453749, −4.13929291666599991459764926166, −2.50072217342111410854292656830, −1.11932982030622035772854814345, 0.62502996856896733153832024092, 3.10190462943463848638649673869, 3.68840967254920117682529245941, 4.58346559615588299095831038465, 5.69127137358146195694214582592, 6.07223032664008355394529099887, 7.39043046538568201890217681819, 8.123229017607925610638215242756, 9.121650881942513946089588063391, 9.753307504869741309727882461104

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