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Properties

Label 2-672-7.4-c1-0-15
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  + (0.5 − 0.866i)3-s + (−1.5 − 2.59i)5-s + (−0.5 − 2.59i)7-s + (−0.499 − 0.866i)9-s + (−0.5 + 0.866i)11-s − 4·13-s − 3·15-s + (−2 + 3.46i)17-s + (−2.5 − 0.866i)21-s + (4 + 6.92i)23-s + (−2 + 3.46i)25-s − 0.999·27-s − 7·29-s + (5.5 − 9.52i)31-s + (0.499 + 0.866i)33-s + ⋯
L(s)  = 1  + (0.288 − 0.499i)3-s + (−0.670 − 1.16i)5-s + (−0.188 − 0.981i)7-s + (−0.166 − 0.288i)9-s + (−0.150 + 0.261i)11-s − 1.10·13-s − 0.774·15-s + (−0.485 + 0.840i)17-s + (−0.545 − 0.188i)21-s + (0.834 + 1.44i)23-s + (−0.400 + 0.692i)25-s − 0.192·27-s − 1.29·29-s + (0.987 − 1.71i)31-s + (0.0870 + 0.150i)33-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: $\chi_{672} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 672,\ (\ :1/2),\ -0.991 + 0.126i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0517900 - 0.816102i\)
\(L(\frac12)\) \(\approx\) \(0.0517900 - 0.816102i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.5 + 2.59i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 4T + 13T^{2} \)
17 \( 1 + (2 - 3.46i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4 - 6.92i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 7T + 29T^{2} \)
31 \( 1 + (-5.5 + 9.52i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2 + 3.46i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 4T + 41T^{2} \)
43 \( 1 - 2T + 43T^{2} \)
47 \( 1 + (1 + 1.73i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.5 + 9.52i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.5 + 6.06i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5 + 8.66i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5 + 8.66i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + (-3 + 5.19i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.5 - 9.52i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 11T + 83T^{2} \)
89 \( 1 + (3 + 5.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 7T + 97T^{2} \)
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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

−9.882724046586111032315442386007, −9.243528528350979765049350673891, −8.142376629791072338346520784747, −7.60918841440392638251361485796, −6.79446746760302646877784681703, −5.41150339553893573471232803649, −4.43565797025948028247464948349, −3.56959191540855915071565191122, −1.88000897355429537210384023582, −0.40301410796786540700582645278, 2.61388335700796175884529320249, 3.03274043102579757558441036901, 4.44874214861656310617675544730, 5.39845658638549212085684967099, 6.69277756068589216549189124943, 7.30564475122720038630997502135, 8.459368790109108018929716642288, 9.143411649362586696831363370051, 10.16770161126474226271911820346, 10.82018412178712492693989784384

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