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Properties

Label 2-672-224.171-c1-0-7
Degree $2$
Conductor $672$
Sign $0.136 - 0.990i$
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  + (−0.852 − 1.12i)2-s + (−0.130 + 0.991i)3-s + (−0.545 + 1.92i)4-s + (−0.0401 − 0.304i)5-s + (1.22 − 0.698i)6-s + (−2.41 − 1.08i)7-s + (2.63 − 1.02i)8-s + (−0.965 − 0.258i)9-s + (−0.309 + 0.305i)10-s + (1.50 + 1.95i)11-s + (−1.83 − 0.792i)12-s + (1.81 − 0.752i)13-s + (0.829 + 3.64i)14-s + 0.307·15-s + (−3.40 − 2.10i)16-s + (−1.47 + 2.55i)17-s + ⋯
L(s)  = 1  + (−0.602 − 0.797i)2-s + (−0.0753 + 0.572i)3-s + (−0.272 + 0.962i)4-s + (−0.0179 − 0.136i)5-s + (0.502 − 0.285i)6-s + (−0.911 − 0.411i)7-s + (0.932 − 0.362i)8-s + (−0.321 − 0.0862i)9-s + (−0.0978 + 0.0964i)10-s + (0.452 + 0.589i)11-s + (−0.530 − 0.228i)12-s + (0.503 − 0.208i)13-s + (0.221 + 0.975i)14-s + 0.0793·15-s + (−0.851 − 0.525i)16-s + (−0.358 + 0.620i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.136 - 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.136 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $0.136 - 0.990i$
Analytic conductor: \(5.36594\)
Root analytic conductor: \(2.31645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{672} (619, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 672,\ (\ :1/2),\ 0.136 - 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.451689 + 0.393698i\)
\(L(\frac12)\) \(\approx\) \(0.451689 + 0.393698i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.852 + 1.12i)T \)
3 \( 1 + (0.130 - 0.991i)T \)
7 \( 1 + (2.41 + 1.08i)T \)
good5 \( 1 + (0.0401 + 0.304i)T + (-4.82 + 1.29i)T^{2} \)
11 \( 1 + (-1.50 - 1.95i)T + (-2.84 + 10.6i)T^{2} \)
13 \( 1 + (-1.81 + 0.752i)T + (9.19 - 9.19i)T^{2} \)
17 \( 1 + (1.47 - 2.55i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.554 + 0.723i)T + (-4.91 - 18.3i)T^{2} \)
23 \( 1 + (1.72 - 6.45i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (6.73 - 2.79i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 + (3.52 - 6.10i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.72 - 0.358i)T + (35.7 - 9.57i)T^{2} \)
41 \( 1 + (-1.44 - 1.44i)T + 41iT^{2} \)
43 \( 1 + (3.13 - 7.56i)T + (-30.4 - 30.4i)T^{2} \)
47 \( 1 + (9.05 - 5.22i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.40 - 3.37i)T + (13.7 - 51.1i)T^{2} \)
59 \( 1 + (-1.20 - 1.56i)T + (-15.2 + 56.9i)T^{2} \)
61 \( 1 + (-3.88 + 5.05i)T + (-15.7 - 58.9i)T^{2} \)
67 \( 1 + (5.43 + 0.715i)T + (64.7 + 17.3i)T^{2} \)
71 \( 1 + (-4.69 + 4.69i)T - 71iT^{2} \)
73 \( 1 + (-3.61 + 0.967i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-4.48 - 7.77i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.70 - 13.7i)T + (-58.6 + 58.6i)T^{2} \)
89 \( 1 + (-11.5 - 3.10i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + 10.8iT - 97T^{2} \)
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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

−10.73774104191306511155914917833, −9.702198456437195664871572421070, −9.354601939905145441448553631455, −8.388353600033417365990542758816, −7.32347532615096272781055970045, −6.39014765729378845572682958151, −4.96577374268538340880779638039, −3.84086831009242366989058145284, −3.19727939404233080248607924321, −1.51372570313127491013637316321, 0.40394845621942775916060397768, 2.14553597872014835146169494792, 3.68970052013716652714569957513, 5.21219381541474886888414203188, 6.19693403122664035582606832344, 6.66526919215403494040462100148, 7.59840437601111426784858783112, 8.680055667007877282603934951295, 9.135924452140603056513587506552, 10.13245102753660515606169965468

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