Properties

Label 2-672-7.4-c1-0-6
Degree $2$
Conductor $672$
Sign $0.968 + 0.250i$
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  + (−0.5 + 0.866i)3-s + (0.5 − 2.59i)7-s + (−0.499 − 0.866i)9-s + (−1 + 1.73i)11-s + 5·13-s + (1 − 1.73i)17-s + (−1.5 − 2.59i)19-s + (2 + 1.73i)21-s + (−1 − 1.73i)23-s + (2.5 − 4.33i)25-s + 0.999·27-s + 8·29-s + (0.5 − 0.866i)31-s + (−0.999 − 1.73i)33-s + (2.5 + 4.33i)37-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + (0.188 − 0.981i)7-s + (−0.166 − 0.288i)9-s + (−0.301 + 0.522i)11-s + 1.38·13-s + (0.242 − 0.420i)17-s + (−0.344 − 0.596i)19-s + (0.436 + 0.377i)21-s + (−0.208 − 0.361i)23-s + (0.5 − 0.866i)25-s + 0.192·27-s + 1.48·29-s + (0.0898 − 0.155i)31-s + (−0.174 − 0.301i)33-s + (0.410 + 0.711i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.250i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.968 + 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $0.968 + 0.250i$
Analytic conductor: \(5.36594\)
Root analytic conductor: \(2.31645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{672} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 672,\ (\ :1/2),\ 0.968 + 0.250i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.40438 - 0.178965i\)
\(L(\frac12)\) \(\approx\) \(1.40438 - 0.178965i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (-0.5 + 2.59i)T \)
good5 \( 1 + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1 - 1.73i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 5T + 13T^{2} \)
17 \( 1 + (-1 + 1.73i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.5 + 2.59i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1 + 1.73i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 8T + 29T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.5 - 4.33i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 - 7T + 43T^{2} \)
47 \( 1 + (-4 - 6.92i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1 + 1.73i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (5 - 8.66i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1 - 1.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.5 + 9.52i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 + (-1.5 + 2.59i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (8.5 + 14.7i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 16T + 83T^{2} \)
89 \( 1 + (6 + 10.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 14T + 97T^{2} \)
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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

−10.58213763554481124016131105448, −9.790521770964003399701452324252, −8.744874791082726011338919275599, −7.916296783857286601235545036002, −6.83672102077519144216207351768, −6.05069579034749864449501904981, −4.72466592405287022646759610789, −4.15373004040385001547008461257, −2.83137774342708755548891080847, −0.956869995153323757616530987507, 1.30865545830965751972136080827, 2.68689873440682736571877829636, 3.93544302651761366684786896053, 5.43432835245337445491537850177, 5.92968103104968451139313600533, 6.89022464677755318365992258003, 8.228142734744287722552979378006, 8.495147862261865503692863698604, 9.633269741072575811828574383452, 10.83664555508901704742747250637

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