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Properties

Label 1-709-709.2-r1-0-0
Degree $1$
Conductor $709$
Sign $0.473 + 0.880i$
Analytic cond. $76.1926$
Root an. cond. $76.1926$
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  + (0.999 + 0.00887i)2-s + (0.0443 + 0.999i)3-s + (0.999 + 0.0177i)4-s + (−0.959 + 0.280i)5-s + (0.0354 + 0.999i)6-s + (0.895 + 0.445i)7-s + (0.999 + 0.0266i)8-s + (−0.996 + 0.0886i)9-s + (−0.962 + 0.271i)10-s + (0.943 − 0.330i)11-s + (0.0266 + 0.999i)12-s + (0.552 − 0.833i)13-s + (0.891 + 0.453i)14-s + (−0.322 − 0.946i)15-s + (0.999 + 0.0354i)16-s + (0.651 + 0.758i)17-s + ⋯
L(s)  = 1  + (0.999 + 0.00887i)2-s + (0.0443 + 0.999i)3-s + (0.999 + 0.0177i)4-s + (−0.959 + 0.280i)5-s + (0.0354 + 0.999i)6-s + (0.895 + 0.445i)7-s + (0.999 + 0.0266i)8-s + (−0.996 + 0.0886i)9-s + (−0.962 + 0.271i)10-s + (0.943 − 0.330i)11-s + (0.0266 + 0.999i)12-s + (0.552 − 0.833i)13-s + (0.891 + 0.453i)14-s + (−0.322 − 0.946i)15-s + (0.999 + 0.0354i)16-s + (0.651 + 0.758i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.473 + 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.473 + 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(709\)
Sign: $0.473 + 0.880i$
Analytic conductor: \(76.1926\)
Root analytic conductor: \(76.1926\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{709} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 709,\ (1:\ ),\ 0.473 + 0.880i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.201300308 + 2.511362699i\)
\(L(\frac12)\) \(\approx\) \(4.201300308 + 2.511362699i\)
\(L(1)\) \(\approx\) \(2.094586779 + 0.7814432218i\)
\(L(1)\) \(\approx\) \(2.094586779 + 0.7814432218i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad709 \( 1 \)
good2 \( 1 + (0.999 + 0.00887i)T \)
3 \( 1 + (0.0443 + 0.999i)T \)
5 \( 1 + (-0.959 + 0.280i)T \)
7 \( 1 + (0.895 + 0.445i)T \)
11 \( 1 + (0.943 - 0.330i)T \)
13 \( 1 + (0.552 - 0.833i)T \)
17 \( 1 + (0.651 + 0.758i)T \)
19 \( 1 + (0.879 - 0.476i)T \)
23 \( 1 + (-0.176 - 0.984i)T \)
29 \( 1 + (0.823 + 0.567i)T \)
31 \( 1 + (-0.194 + 0.981i)T \)
37 \( 1 + (0.445 - 0.895i)T \)
41 \( 1 + (-0.775 - 0.631i)T \)
43 \( 1 + (0.372 - 0.928i)T \)
47 \( 1 + (0.339 - 0.940i)T \)
53 \( 1 + (0.752 + 0.658i)T \)
59 \( 1 + (-0.0266 + 0.999i)T \)
61 \( 1 + (-0.988 + 0.150i)T \)
67 \( 1 + (0.684 + 0.728i)T \)
71 \( 1 + (-0.962 - 0.271i)T \)
73 \( 1 + (0.638 - 0.769i)T \)
79 \( 1 + (-0.995 - 0.0974i)T \)
83 \( 1 + (-0.921 + 0.388i)T \)
89 \( 1 + (0.0709 + 0.997i)T \)
97 \( 1 + (0.703 + 0.710i)T \)
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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

−22.68066262597399142421289000097, −21.4236319770501384137886465554, −20.46160181915038470588090411882, −20.06790943273036126244340003901, −19.18683182014271890099852483221, −18.37331537383051171042959014406, −17.16971606457593629944862000731, −16.54175427485431869936310285807, −15.54559083523213205312035721666, −14.49049526503747648443804175743, −14.06021818481310051187091302894, −13.2422887175043746224834537193, −12.13784556458107462576853999166, −11.55557286240879648557350667949, −11.35022141056921801360659767196, −9.645512241486720887358766340658, −8.25688818214447434859268014921, −7.61460865950831032672769433825, −6.95002768356082045391774440799, −5.944372972786259947237997797334, −4.8243978818563717116442316567, −4.00573079335382486828835516561, −3.0664713223881118767904590768, −1.608934244053957093673494817625, −1.04601955623602533155345612789, 1.05848016204599725213305279017, 2.6799448211763875810697066762, 3.538069093514857728355685781603, 4.181453227741441557763584690684, 5.15446752550953358846057475192, 5.88952563337407168244535268800, 7.10393178606230135261359648816, 8.2055663434057776753569081694, 8.81830378409718871990639447948, 10.534300645785471446289866048355, 10.79919467820157675053372415441, 11.914342328283273671657373693230, 12.19534617572721484645312126337, 13.77984234443578792853976480579, 14.536019694254901050216389188796, 15.012037948367309686686475445681, 15.79221733928374655433265857043, 16.41837452706921234369923488104, 17.421102144529315616383302380248, 18.60483252142623940044975819131, 19.853946772934971370691682849114, 20.11864272157115481869112558330, 21.1392442271643209308965404282, 21.81236271461096048797705002455, 22.45610124936376832238842687705

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