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Properties

Label 2-672-224.195-c1-0-19
Degree $2$
Conductor $672$
Sign $0.994 + 0.100i$
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  + (0.000203 + 1.41i)2-s + (−0.923 − 0.382i)3-s + (−1.99 + 0.000576i)4-s + (−2.85 + 1.18i)5-s + (0.541 − 1.30i)6-s + (−2.33 + 1.23i)7-s + (−0.00122 − 2.82i)8-s + (0.707 + 0.707i)9-s + (−1.67 − 4.03i)10-s + (−0.888 − 2.14i)11-s + (1.84 + 0.764i)12-s + (0.969 + 0.401i)13-s + (−1.75 − 3.30i)14-s + 3.08·15-s + (3.99 − 0.00230i)16-s + 1.29·17-s + ⋯
L(s)  = 1  + (0.000144 + 0.999i)2-s + (−0.533 − 0.220i)3-s + (−0.999 + 0.000288i)4-s + (−1.27 + 0.528i)5-s + (0.220 − 0.533i)6-s + (−0.883 + 0.468i)7-s + (−0.000432 − 0.999i)8-s + (0.235 + 0.235i)9-s + (−0.528 − 1.27i)10-s + (−0.267 − 0.646i)11-s + (0.533 + 0.220i)12-s + (0.269 + 0.111i)13-s + (−0.468 − 0.883i)14-s + 0.797·15-s + (0.999 − 0.000576i)16-s + 0.313·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.100i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 + 0.100i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $0.994 + 0.100i$
Analytic conductor: \(5.36594\)
Root analytic conductor: \(2.31645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{672} (643, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 672,\ (\ :1/2),\ 0.994 + 0.100i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.460451 - 0.0231812i\)
\(L(\frac12)\) \(\approx\) \(0.460451 - 0.0231812i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.000203 - 1.41i)T \)
3 \( 1 + (0.923 + 0.382i)T \)
7 \( 1 + (2.33 - 1.23i)T \)
good5 \( 1 + (2.85 - 1.18i)T + (3.53 - 3.53i)T^{2} \)
11 \( 1 + (0.888 + 2.14i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 + (-0.969 - 0.401i)T + (9.19 + 9.19i)T^{2} \)
17 \( 1 - 1.29T + 17T^{2} \)
19 \( 1 + (0.848 - 2.04i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (-4.63 + 4.63i)T - 23iT^{2} \)
29 \( 1 + (1.01 + 0.419i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 + 2.00T + 31T^{2} \)
37 \( 1 + (-2.86 - 6.92i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (-8.74 + 8.74i)T - 41iT^{2} \)
43 \( 1 + (-0.264 - 0.639i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 + 7.54iT - 47T^{2} \)
53 \( 1 + (7.48 - 3.09i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (5.41 + 13.0i)T + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (-0.699 + 1.68i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 + (-3.75 + 9.07i)T + (-47.3 - 47.3i)T^{2} \)
71 \( 1 + (-7.41 - 7.41i)T + 71iT^{2} \)
73 \( 1 + (-1.19 + 1.19i)T - 73iT^{2} \)
79 \( 1 + 8.25T + 79T^{2} \)
83 \( 1 + (-1.34 + 3.24i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (2.93 + 2.93i)T + 89iT^{2} \)
97 \( 1 - 9.13iT - 97T^{2} \)
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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

−10.55344770684820090528157803971, −9.455009087612980177357681973486, −8.478128882306954333304082701496, −7.76758898850420467256951789656, −6.85333855505086551430383413881, −6.23184562355642354151595121957, −5.25493344638920951517099818174, −4.02950064493436293506698090567, −3.12899792853022250471537216116, −0.35955860830438614174870808259, 0.971938689405859360487572199060, 2.99552275345663773337516025422, 3.98936327284052694947326765208, 4.62949676943102800573717437402, 5.75741507481298582130836680342, 7.20424531271289595013503061444, 7.943008992359608406086011250496, 9.161043848268634530975459713949, 9.678015722957199779522619409931, 10.80830182484976456023093132851

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