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Properties

Label 2-693-77.48-c0-0-0
Degree $2$
Conductor $693$
Sign $0.0237 + 0.999i$
Analytic cond. $0.345852$
Root an. cond. $0.588091$
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  + (−1.53 − 1.11i)2-s + (0.809 + 2.48i)4-s + (−0.309 − 0.951i)7-s + (0.951 − 2.92i)8-s + (0.587 + 0.809i)11-s + (−0.587 + 1.80i)14-s + (−2.61 + 1.90i)16-s − 1.90i·22-s + 1.90·23-s + (0.309 − 0.951i)25-s + (2.11 − 1.53i)28-s + (−0.363 − 1.11i)29-s + 3.07·32-s + (−0.190 − 0.587i)37-s + 1.61·43-s + (−1.53 + 2.11i)44-s + ⋯
L(s)  = 1  + (−1.53 − 1.11i)2-s + (0.809 + 2.48i)4-s + (−0.309 − 0.951i)7-s + (0.951 − 2.92i)8-s + (0.587 + 0.809i)11-s + (−0.587 + 1.80i)14-s + (−2.61 + 1.90i)16-s − 1.90i·22-s + 1.90·23-s + (0.309 − 0.951i)25-s + (2.11 − 1.53i)28-s + (−0.363 − 1.11i)29-s + 3.07·32-s + (−0.190 − 0.587i)37-s + 1.61·43-s + (−1.53 + 2.11i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0237 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0237 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $0.0237 + 0.999i$
Analytic conductor: \(0.345852\)
Root analytic conductor: \(0.588091\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{693} (433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 693,\ (\ :0),\ 0.0237 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4603585022\)
\(L(\frac12)\) \(\approx\) \(0.4603585022\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (0.309 + 0.951i)T \)
11 \( 1 + (-0.587 - 0.809i)T \)
good2 \( 1 + (1.53 + 1.11i)T + (0.309 + 0.951i)T^{2} \)
5 \( 1 + (-0.309 + 0.951i)T^{2} \)
13 \( 1 + (-0.309 - 0.951i)T^{2} \)
17 \( 1 + (-0.309 + 0.951i)T^{2} \)
19 \( 1 + (0.809 + 0.587i)T^{2} \)
23 \( 1 - 1.90T + T^{2} \)
29 \( 1 + (0.363 + 1.11i)T + (-0.809 + 0.587i)T^{2} \)
31 \( 1 + (-0.309 - 0.951i)T^{2} \)
37 \( 1 + (0.190 + 0.587i)T + (-0.809 + 0.587i)T^{2} \)
41 \( 1 + (0.809 + 0.587i)T^{2} \)
43 \( 1 - 1.61T + T^{2} \)
47 \( 1 + (0.809 + 0.587i)T^{2} \)
53 \( 1 + (0.951 + 0.690i)T + (0.309 + 0.951i)T^{2} \)
59 \( 1 + (0.809 - 0.587i)T^{2} \)
61 \( 1 + (-0.309 + 0.951i)T^{2} \)
67 \( 1 + 1.61T + T^{2} \)
71 \( 1 + (-0.951 + 0.690i)T + (0.309 - 0.951i)T^{2} \)
73 \( 1 + (0.809 - 0.587i)T^{2} \)
79 \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \)
83 \( 1 + (-0.309 + 0.951i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-0.309 - 0.951i)T^{2} \)
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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

−10.50020098619143192761601827958, −9.534521051531393810509189631233, −9.156423215927911995231321270030, −8.006120843950209007586739503307, −7.28409487718031471102412253892, −6.55842952760049731892465417006, −4.50254563103778802406453216618, −3.54609656601520768809208747270, −2.37118010573218337153350410883, −1.00183530503990345042120480982, 1.34739399174941938019478509485, 3.00403653044493241788977459216, 5.08214499233815258295364921669, 5.85018459067842607722955738099, 6.67586411438720131305083836267, 7.42602950802704309884997689260, 8.514104180368091805562152184800, 9.072958959368250335261617002486, 9.488302408199264340459984915077, 10.79376272061957872363800837253

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