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Properties

Label 2-672-168.5-c1-0-0
Degree $2$
Conductor $672$
Sign $-0.935 + 0.352i$
Analytic cond. $5.36594$
Root an. cond. $2.31645$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.0574 + 1.73i)3-s + (−0.461 − 0.266i)5-s + (−0.489 + 2.60i)7-s + (−2.99 − 0.198i)9-s + (−2.28 − 3.96i)11-s − 4.97·13-s + (0.487 − 0.783i)15-s + (2.16 + 3.74i)17-s + (−0.921 + 1.59i)19-s + (−4.47 − 0.996i)21-s + (0.103 + 0.0596i)23-s + (−2.35 − 4.08i)25-s + (0.516 − 5.17i)27-s − 7.74·29-s + (1.93 − 1.11i)31-s + ⋯
L(s)  = 1  + (−0.0331 + 0.999i)3-s + (−0.206 − 0.119i)5-s + (−0.184 + 0.982i)7-s + (−0.997 − 0.0663i)9-s + (−0.689 − 1.19i)11-s − 1.38·13-s + (0.125 − 0.202i)15-s + (0.524 + 0.907i)17-s + (−0.211 + 0.366i)19-s + (−0.976 − 0.217i)21-s + (0.0215 + 0.0124i)23-s + (−0.471 − 0.816i)25-s + (0.0993 − 0.995i)27-s − 1.43·29-s + (0.347 − 0.200i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.935 + 0.352i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.935 + 0.352i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $-0.935 + 0.352i$
Analytic conductor: \(5.36594\)
Root analytic conductor: \(2.31645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{672} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 672,\ (\ :1/2),\ -0.935 + 0.352i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0648465 - 0.356501i\)
\(L(\frac12)\) \(\approx\) \(0.0648465 - 0.356501i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.0574 - 1.73i)T \)
7 \( 1 + (0.489 - 2.60i)T \)
good5 \( 1 + (0.461 + 0.266i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.28 + 3.96i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 4.97T + 13T^{2} \)
17 \( 1 + (-2.16 - 3.74i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.921 - 1.59i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.103 - 0.0596i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 7.74T + 29T^{2} \)
31 \( 1 + (-1.93 + 1.11i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-7.02 - 4.05i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 2.60T + 41T^{2} \)
43 \( 1 - 1.87iT - 43T^{2} \)
47 \( 1 + (2.91 - 5.04i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.29 + 3.97i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.71 - 3.29i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.07 + 1.86i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (10.4 - 6.05i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 6.20iT - 71T^{2} \)
73 \( 1 + (-8.35 + 4.82i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.0228 + 0.0396i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 3.86iT - 83T^{2} \)
89 \( 1 + (8.23 - 14.2i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 7.18iT - 97T^{2} \)
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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

−10.92825257252751336739316249321, −10.01652269875224809675960587479, −9.414002003788811816358239928725, −8.381680455104826467183196305635, −7.87278717915983226079560546444, −6.15614733491609762039044477969, −5.58112291860665797460927796402, −4.61735108195686012467895803282, −3.42206615729774019757882211948, −2.47113651794719438953149749697, 0.17857840786918425117994106252, 1.94242112147328107928365626382, 3.08104675766871970793609086246, 4.54863786838190604664205582909, 5.46757028769771869791614914206, 6.83370445063047904396869185184, 7.48210462595618130658788615535, 7.72917632776660125418568361168, 9.306748473548165413429396834542, 9.962279656456350349997604717601

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