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Properties

Label 2-1407-1407.485-c0-0-0
Degree $2$
Conductor $1407$
Sign $-0.360 + 0.932i$
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  + (0.580 − 0.814i)3-s + (−0.142 − 0.989i)4-s + (0.0475 − 0.998i)7-s + (−0.327 − 0.945i)9-s + (−0.888 − 0.458i)12-s + (1.16 + 0.600i)13-s + (−0.959 + 0.281i)16-s + (1.30 + 1.50i)19-s + (−0.786 − 0.618i)21-s + (−0.888 − 0.458i)25-s + (−0.959 − 0.281i)27-s + (−0.995 + 0.0950i)28-s + (−1.61 − 1.03i)31-s + (−0.888 + 0.458i)36-s + 0.0951·37-s + ⋯
L(s)  = 1  + (0.580 − 0.814i)3-s + (−0.142 − 0.989i)4-s + (0.0475 − 0.998i)7-s + (−0.327 − 0.945i)9-s + (−0.888 − 0.458i)12-s + (1.16 + 0.600i)13-s + (−0.959 + 0.281i)16-s + (1.30 + 1.50i)19-s + (−0.786 − 0.618i)21-s + (−0.888 − 0.458i)25-s + (−0.959 − 0.281i)27-s + (−0.995 + 0.0950i)28-s + (−1.61 − 1.03i)31-s + (−0.888 + 0.458i)36-s + 0.0951·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.360 + 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.360 + 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1407\)    =    \(3 \cdot 7 \cdot 67\)
Sign: $-0.360 + 0.932i$
Analytic conductor: \(0.702184\)
Root analytic conductor: \(0.837964\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1407} (485, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1407,\ (\ :0),\ -0.360 + 0.932i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.285023674\)
\(L(\frac12)\) \(\approx\) \(1.285023674\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.580 + 0.814i)T \)
7 \( 1 + (-0.0475 + 0.998i)T \)
67 \( 1 + (-0.0475 - 0.998i)T \)
good2 \( 1 + (0.142 + 0.989i)T^{2} \)
5 \( 1 + (0.888 + 0.458i)T^{2} \)
11 \( 1 + (-0.841 + 0.540i)T^{2} \)
13 \( 1 + (-1.16 - 0.600i)T + (0.580 + 0.814i)T^{2} \)
17 \( 1 + (-0.235 - 0.971i)T^{2} \)
19 \( 1 + (-1.30 - 1.50i)T + (-0.142 + 0.989i)T^{2} \)
23 \( 1 + (0.654 - 0.755i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (1.61 + 1.03i)T + (0.415 + 0.909i)T^{2} \)
37 \( 1 - 0.0951T + T^{2} \)
41 \( 1 + (-0.235 - 0.971i)T^{2} \)
43 \( 1 + (0.239 - 1.66i)T + (-0.959 - 0.281i)T^{2} \)
47 \( 1 + (0.327 + 0.945i)T^{2} \)
53 \( 1 + (-0.235 + 0.971i)T^{2} \)
59 \( 1 + (0.995 + 0.0950i)T^{2} \)
61 \( 1 + (-1.91 - 0.560i)T + (0.841 + 0.540i)T^{2} \)
71 \( 1 + (-0.723 - 0.690i)T^{2} \)
73 \( 1 + (-0.341 + 0.325i)T + (0.0475 - 0.998i)T^{2} \)
79 \( 1 + (0.0845 + 0.0436i)T + (0.580 + 0.814i)T^{2} \)
83 \( 1 + (-0.0475 - 0.998i)T^{2} \)
89 \( 1 + (0.327 - 0.945i)T^{2} \)
97 \( 1 + (0.580 + 1.00i)T + (-0.5 + 0.866i)T^{2} \)
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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

−9.648631230517813868757717981988, −8.658769107386455412488638748974, −7.83147575286239518902118746438, −7.12644013386122742599449131838, −6.19493265325038044950690604779, −5.64747770791322340062508034673, −4.19561138777472687884589442303, −3.50797120721916727971454712935, −1.90031333779131468472251436522, −1.11157304428196268347825181750, 2.19750750323804230692224277781, 3.24414036292661692080841957735, 3.70184691778486151517826177266, 5.01271695067920965611599707026, 5.56125818435021934644074266136, 6.95820982358347898647814618512, 7.80952661452520594933479267285, 8.607274123934965718670109438201, 9.028497610136163522443797177422, 9.702473992337002380836526161567

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