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Properties

Label 2-672-21.5-c1-0-12
Degree $2$
Conductor $672$
Sign $0.991 + 0.126i$
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  + (−0.866 − 1.5i)3-s + (−1.32 + 2.29i)5-s + (2.29 − 1.32i)7-s + (−1.5 + 2.59i)9-s + (−0.866 + 0.5i)11-s − 3.46i·13-s + 4.58·15-s + (2.64 + 4.58i)17-s + (4.58 + 2.64i)19-s + (−3.96 − 2.29i)21-s + (1.73 + i)23-s + (−1 − 1.73i)25-s + 5.19·27-s − 4.58i·29-s + (6.87 − 3.96i)31-s + ⋯
L(s)  = 1  + (−0.499 − 0.866i)3-s + (−0.591 + 1.02i)5-s + (0.866 − 0.499i)7-s + (−0.5 + 0.866i)9-s + (−0.261 + 0.150i)11-s − 0.960i·13-s + 1.18·15-s + (0.641 + 1.11i)17-s + (1.05 + 0.606i)19-s + (−0.866 − 0.500i)21-s + (0.361 + 0.208i)23-s + (−0.200 − 0.346i)25-s + 1.00·27-s − 0.850i·29-s + (1.23 − 0.712i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $0.991 + 0.126i$
Analytic conductor: \(5.36594\)
Root analytic conductor: \(2.31645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{672} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 672,\ (\ :1/2),\ 0.991 + 0.126i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.23832 - 0.0785841i\)
\(L(\frac12)\) \(\approx\) \(1.23832 - 0.0785841i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 + 1.5i)T \)
7 \( 1 + (-2.29 + 1.32i)T \)
good5 \( 1 + (1.32 - 2.29i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.866 - 0.5i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 3.46iT - 13T^{2} \)
17 \( 1 + (-2.64 - 4.58i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.58 - 2.64i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.73 - i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 4.58iT - 29T^{2} \)
31 \( 1 + (-6.87 + 3.96i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (3 - 5.19i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 10.5T + 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + (-3.46 + 6i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.96 - 2.29i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.06 + 10.5i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-12 - 6.92i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.58 - 7.93i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 4iT - 71T^{2} \)
73 \( 1 + (6 - 3.46i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.29 - 3.96i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 1.73T + 83T^{2} \)
89 \( 1 + (2.64 - 4.58i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 8.66iT - 97T^{2} \)
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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

−10.62601129600572412760844432426, −10.01497941286463823377726670397, −8.167375356256971822704097905441, −7.84861006244785286832304709117, −7.13493541977314768136236677908, −6.07386963284428799903782780652, −5.20720435107478518313825418159, −3.83922101874522064698186962343, −2.64307183575891835413264397995, −1.10699186861306373941749029936, 0.958659548847434545362513805522, 2.95086395418404648738431891695, 4.37600914229892118006587994266, 4.91561507610999597154916023165, 5.63072805536699132908888149997, 7.04642266516502734788713202217, 8.093989907080329236785504553659, 9.030678660540712306882318682271, 9.388694553312725847188069981547, 10.67420239231442029133504563128

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