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Properties

Label 2-672-168.125-c1-0-4
Degree $2$
Conductor $672$
Sign $-0.925 - 0.377i$
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  + 1.73i·3-s + 3.46i·5-s + (−1 + 2.44i)7-s − 2.99·9-s + 5.65·11-s − 5.99·15-s + (−4.24 − 1.73i)21-s − 6.99·25-s − 5.19i·27-s − 2.82·29-s − 4.89i·31-s + 9.79i·33-s + (−8.48 − 3.46i)35-s − 10.3i·45-s + (−4.99 − 4.89i)49-s + ⋯
L(s)  = 1  + 0.999i·3-s + 1.54i·5-s + (−0.377 + 0.925i)7-s − 0.999·9-s + 1.70·11-s − 1.54·15-s + (−0.925 − 0.377i)21-s − 1.39·25-s − 0.999i·27-s − 0.525·29-s − 0.879i·31-s + 1.70i·33-s + (−1.43 − 0.585i)35-s − 1.54i·45-s + (−0.714 − 0.699i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.925 - 0.377i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.925 - 0.377i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $-0.925 - 0.377i$
Analytic conductor: \(5.36594\)
Root analytic conductor: \(2.31645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{672} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 672,\ (\ :1/2),\ -0.925 - 0.377i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.262541 + 1.33771i\)
\(L(\frac12)\) \(\approx\) \(0.262541 + 1.33771i\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.73iT \)
7 \( 1 + (1 - 2.44i)T \)
good5 \( 1 - 3.46iT - 5T^{2} \)
11 \( 1 - 5.65T + 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + 2.82T + 29T^{2} \)
31 \( 1 + 4.89iT - 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 14.1T + 53T^{2} \)
59 \( 1 - 10.3iT - 59T^{2} \)
61 \( 1 + 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + 9.79iT - 73T^{2} \)
79 \( 1 + 10T + 79T^{2} \)
83 \( 1 - 17.3iT - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 19.5iT - 97T^{2} \)
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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

−10.87572108290081376913891838146, −9.954386747450927637350974507079, −9.344433078952771707809154979347, −8.558429024716968860495106574845, −7.18924088271140654301828127326, −6.32208343310349744768570590630, −5.67082149846391749652466834147, −4.14857666032439485655813246039, −3.37042060032065511476602263885, −2.37420911996708029517998665191, 0.77161237921251292897969993807, 1.64788377480078058062671648637, 3.56616294633269589935088169911, 4.53851117458343401867218624076, 5.70988531001012266044390236368, 6.67228420866595203106852184293, 7.41694071745338381504248610325, 8.530437724258779464107403466682, 9.014256551683698170878404700087, 9.946314582995527364632722666832

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