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Properties

Label 1-684-684.463-r1-0-0
Degree $1$
Conductor $684$
Sign $0.846 - 0.532i$
Analytic cond. $73.5060$
Root an. cond. $73.5060$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s + (0.5 − 0.866i)7-s + (0.5 − 0.866i)11-s + (−0.5 + 0.866i)13-s + (−0.5 + 0.866i)17-s + (0.5 − 0.866i)23-s + 25-s + 29-s + (0.5 + 0.866i)31-s + (0.5 − 0.866i)35-s + 37-s + 41-s + (0.5 + 0.866i)43-s − 47-s + (−0.5 − 0.866i)49-s + ⋯
L(s)  = 1  + 5-s + (0.5 − 0.866i)7-s + (0.5 − 0.866i)11-s + (−0.5 + 0.866i)13-s + (−0.5 + 0.866i)17-s + (0.5 − 0.866i)23-s + 25-s + 29-s + (0.5 + 0.866i)31-s + (0.5 − 0.866i)35-s + 37-s + 41-s + (0.5 + 0.866i)43-s − 47-s + (−0.5 − 0.866i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.846 - 0.532i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.846 - 0.532i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $0.846 - 0.532i$
Analytic conductor: \(73.5060\)
Root analytic conductor: \(73.5060\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (463, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 684,\ (1:\ ),\ 0.846 - 0.532i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.848775231 - 0.8221380036i\)
\(L(\frac12)\) \(\approx\) \(2.848775231 - 0.8221380036i\)
\(L(1)\) \(\approx\) \(1.450728224 - 0.1758296256i\)
\(L(1)\) \(\approx\) \(1.450728224 - 0.1758296256i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
19 \( 1 \)
good5 \( 1 + T \)
7 \( 1 + (0.5 - 0.866i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (-0.5 + 0.866i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + T \)
41 \( 1 + T \)
43 \( 1 + (0.5 + 0.866i)T \)
47 \( 1 - T \)
53 \( 1 + (-0.5 - 0.866i)T \)
59 \( 1 - T \)
61 \( 1 + T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (0.5 - 0.866i)T \)
73 \( 1 + (-0.5 + 0.866i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (0.5 - 0.866i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (-0.5 - 0.866i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−22.368821875623334462364757616971, −21.87350155278107681953686993991, −20.95249156118965723720744345170, −20.30538437985524064670018943796, −19.32255796516199247690967318309, −18.2477329611676666601467259300, −17.66906168046772906082223126911, −17.17070968543404648784551485071, −15.86730347633756462908363290307, −15.07107284773984093646855879553, −14.390124479798006307383864180826, −13.433124209028127568183032703949, −12.58586653269952301477858068434, −11.78351202563311012859172107010, −10.81019540929559896964580970737, −9.65632672294501224166119442365, −9.30681597272494331155413191930, −8.13551879287231838712425232612, −7.12590287178423765409950402993, −6.1015574164077123675747456510, −5.2546874689846602635301539841, −4.507141909203743677743125469713, −2.83313474472910008964589084194, −2.19254769971008421578984285209, −1.008874020413559896526907426983, 0.82644085492920568633992781954, 1.75442637849394177939950424970, 2.9012793307928001630896793007, 4.21219605183049178819451595814, 4.94624517546478552313766347239, 6.29395661954977736710191823463, 6.70187159849086809045502231632, 8.04322568827029791798111691383, 8.88905827570869118672141558724, 9.795645142202882362799646173964, 10.701243910256897434929423238747, 11.33805817890235811750680068767, 12.56945614057481737241355649495, 13.404960706718080942851182808778, 14.29943409668940511890014124915, 14.54046951019936488181691496418, 16.09167705001707640085086277519, 16.8517551627183429170466247515, 17.40050661632036891104418254584, 18.20952871949430595503850428268, 19.30999070454715317631799554268, 19.90115889233163141077017459581, 21.16526091991830410765660897753, 21.38018992268689572180926469447, 22.32053159747362470710831223323

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