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Properties

Label 2-672-224.187-c1-0-9
Degree $2$
Conductor $672$
Sign $0.606 - 0.794i$
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  + (0.661 − 1.25i)2-s + (−0.608 + 0.793i)3-s + (−1.12 − 1.65i)4-s + (2.57 + 3.35i)5-s + (0.589 + 1.28i)6-s + (−1.85 + 1.88i)7-s + (−2.81 + 0.312i)8-s + (−0.258 − 0.965i)9-s + (5.89 − 0.999i)10-s + (−0.360 − 2.73i)11-s + (1.99 + 0.113i)12-s + (−0.167 − 0.404i)13-s + (1.13 + 3.56i)14-s − 4.22·15-s + (−1.46 + 3.72i)16-s + (1.71 + 2.97i)17-s + ⋯
L(s)  = 1  + (0.467 − 0.883i)2-s + (−0.351 + 0.458i)3-s + (−0.562 − 0.826i)4-s + (1.15 + 1.50i)5-s + (0.240 + 0.524i)6-s + (−0.700 + 0.713i)7-s + (−0.993 + 0.110i)8-s + (−0.0862 − 0.321i)9-s + (1.86 − 0.315i)10-s + (−0.108 − 0.825i)11-s + (0.576 + 0.0328i)12-s + (−0.0465 − 0.112i)13-s + (0.303 + 0.952i)14-s − 1.09·15-s + (−0.366 + 0.930i)16-s + (0.416 + 0.720i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.606 - 0.794i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.606 - 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $0.606 - 0.794i$
Analytic conductor: \(5.36594\)
Root analytic conductor: \(2.31645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{672} (187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 672,\ (\ :1/2),\ 0.606 - 0.794i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.35734 + 0.671293i\)
\(L(\frac12)\) \(\approx\) \(1.35734 + 0.671293i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.661 + 1.25i)T \)
3 \( 1 + (0.608 - 0.793i)T \)
7 \( 1 + (1.85 - 1.88i)T \)
good5 \( 1 + (-2.57 - 3.35i)T + (-1.29 + 4.82i)T^{2} \)
11 \( 1 + (0.360 + 2.73i)T + (-10.6 + 2.84i)T^{2} \)
13 \( 1 + (0.167 + 0.404i)T + (-9.19 + 9.19i)T^{2} \)
17 \( 1 + (-1.71 - 2.97i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.986 - 7.49i)T + (-18.3 - 4.91i)T^{2} \)
23 \( 1 + (1.87 - 0.501i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (-0.782 - 1.88i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 + (-1.78 - 3.08i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.97 + 4.58i)T + (9.57 - 35.7i)T^{2} \)
41 \( 1 + (-4.37 - 4.37i)T + 41iT^{2} \)
43 \( 1 + (6.09 + 2.52i)T + (30.4 + 30.4i)T^{2} \)
47 \( 1 + (0.437 + 0.252i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.98 - 0.656i)T + (51.1 - 13.7i)T^{2} \)
59 \( 1 + (0.723 + 5.49i)T + (-56.9 + 15.2i)T^{2} \)
61 \( 1 + (-1.38 + 10.4i)T + (-58.9 - 15.7i)T^{2} \)
67 \( 1 + (-7.15 - 5.48i)T + (17.3 + 64.7i)T^{2} \)
71 \( 1 + (-7.37 + 7.37i)T - 71iT^{2} \)
73 \( 1 + (0.644 - 2.40i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-5.96 + 10.3i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (10.5 - 4.38i)T + (58.6 - 58.6i)T^{2} \)
89 \( 1 + (-4.12 - 15.3i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + 10.9iT - 97T^{2} \)
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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

−10.63989653004954346414280270666, −9.934676655945529157903576129874, −9.522957425369217285461922125490, −8.249555049948124304643906344287, −6.47258486794221599819588029712, −6.02367453441859923140441049905, −5.41043246365811851661293887234, −3.66195689734984777328406238185, −3.05669402001323641098805583150, −1.92033992003960699773166348027, 0.72040936156385213826277007146, 2.52561421467536950939342681367, 4.39482160876591049882818002134, 4.93348435443973556363157884025, 5.92315844790455184129133530867, 6.68783144128706393463490022923, 7.54364608320151712459305920818, 8.588711695361056967532280876055, 9.496940918425584768873791283002, 9.931860768366834393740906995395

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