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Properties

Label 2-672-168.101-c1-0-8
Degree $2$
Conductor $672$
Sign $-0.717 - 0.696i$
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  + (1.11 + 1.32i)3-s + (−0.337 + 0.195i)5-s + (−1.39 + 2.24i)7-s + (−0.529 + 2.95i)9-s + (−0.748 + 1.29i)11-s − 3.28·13-s + (−0.634 − 0.232i)15-s + (−1.68 + 2.91i)17-s + (−2.56 − 4.43i)19-s + (−4.53 + 0.644i)21-s + (4.72 − 2.72i)23-s + (−2.42 + 4.19i)25-s + (−4.51 + 2.57i)27-s + 4.13·29-s + (3.60 + 2.07i)31-s + ⋯
L(s)  = 1  + (0.641 + 0.766i)3-s + (−0.151 + 0.0872i)5-s + (−0.527 + 0.849i)7-s + (−0.176 + 0.984i)9-s + (−0.225 + 0.390i)11-s − 0.911·13-s + (−0.163 − 0.0599i)15-s + (−0.407 + 0.706i)17-s + (−0.587 − 1.01i)19-s + (−0.990 + 0.140i)21-s + (0.985 − 0.569i)23-s + (−0.484 + 0.839i)25-s + (−0.868 + 0.496i)27-s + 0.768·29-s + (0.647 + 0.373i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.717 - 0.696i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.717 - 0.696i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $-0.717 - 0.696i$
Analytic conductor: \(5.36594\)
Root analytic conductor: \(2.31645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{672} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 672,\ (\ :1/2),\ -0.717 - 0.696i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.478814 + 1.18000i\)
\(L(\frac12)\) \(\approx\) \(0.478814 + 1.18000i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.11 - 1.32i)T \)
7 \( 1 + (1.39 - 2.24i)T \)
good5 \( 1 + (0.337 - 0.195i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.748 - 1.29i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 3.28T + 13T^{2} \)
17 \( 1 + (1.68 - 2.91i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.56 + 4.43i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.72 + 2.72i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 4.13T + 29T^{2} \)
31 \( 1 + (-3.60 - 2.07i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (7.46 - 4.31i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 11.1T + 41T^{2} \)
43 \( 1 - 4.79iT - 43T^{2} \)
47 \( 1 + (-2.51 - 4.34i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.499 + 0.864i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.36 - 0.785i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.40 + 5.90i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.05 - 1.76i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 14.3iT - 71T^{2} \)
73 \( 1 + (-2.76 - 1.59i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.239 - 0.414i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 17.4iT - 83T^{2} \)
89 \( 1 + (-2.54 - 4.41i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 9.00iT - 97T^{2} \)
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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

−10.66253444207033854539731906256, −9.881772683117850878383900254790, −9.089893328636361745816745656239, −8.501198947648125227835323295256, −7.41513320550326167354788069733, −6.41235047479388862970421914628, −5.14791443834253780215083503719, −4.42292563342597114944150135603, −3.07457077436700973495587257727, −2.32073368091582759732524139378, 0.60003615748270999845281250319, 2.29036699409534906223808565950, 3.38108660386931422795155050561, 4.43748096860633842391366487970, 5.86459067761015677074710614386, 6.87739989982766047567497901780, 7.48681653318604124910326881743, 8.332732251954475961145544325523, 9.275603629622326439887150268027, 10.07662695478394794552274876867

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