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Properties

Label 1-709-709.201-r0-0-0
Degree $1$
Conductor $709$
Sign $-0.944 + 0.328i$
Analytic cond. $3.29258$
Root an. cond. $3.29258$
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  + (−0.697 + 0.716i)2-s + (0.658 + 0.752i)3-s + (−0.0266 − 0.999i)4-s + (0.910 + 0.413i)5-s + (−0.998 − 0.0532i)6-s + (−0.769 + 0.638i)7-s + (0.734 + 0.678i)8-s + (−0.132 + 0.991i)9-s + (−0.931 + 0.364i)10-s + (0.484 − 0.874i)11-s + (0.734 − 0.678i)12-s + (−0.769 − 0.638i)13-s + (0.0797 − 0.996i)14-s + (0.288 + 0.957i)15-s + (−0.998 + 0.0532i)16-s + (0.484 + 0.874i)17-s + ⋯
L(s)  = 1  + (−0.697 + 0.716i)2-s + (0.658 + 0.752i)3-s + (−0.0266 − 0.999i)4-s + (0.910 + 0.413i)5-s + (−0.998 − 0.0532i)6-s + (−0.769 + 0.638i)7-s + (0.734 + 0.678i)8-s + (−0.132 + 0.991i)9-s + (−0.931 + 0.364i)10-s + (0.484 − 0.874i)11-s + (0.734 − 0.678i)12-s + (−0.769 − 0.638i)13-s + (0.0797 − 0.996i)14-s + (0.288 + 0.957i)15-s + (−0.998 + 0.0532i)16-s + (0.484 + 0.874i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.944 + 0.328i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.944 + 0.328i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(709\)
Sign: $-0.944 + 0.328i$
Analytic conductor: \(3.29258\)
Root analytic conductor: \(3.29258\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{709} (201, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 709,\ (0:\ ),\ -0.944 + 0.328i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1970584688 + 1.166612179i\)
\(L(\frac12)\) \(\approx\) \(0.1970584688 + 1.166612179i\)
\(L(1)\) \(\approx\) \(0.7016454628 + 0.6685728018i\)
\(L(1)\) \(\approx\) \(0.7016454628 + 0.6685728018i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad709 \( 1 \)
good2 \( 1 + (-0.697 + 0.716i)T \)
3 \( 1 + (0.658 + 0.752i)T \)
5 \( 1 + (0.910 + 0.413i)T \)
7 \( 1 + (-0.769 + 0.638i)T \)
11 \( 1 + (0.484 - 0.874i)T \)
13 \( 1 + (-0.769 - 0.638i)T \)
17 \( 1 + (0.484 + 0.874i)T \)
19 \( 1 + (0.734 + 0.678i)T \)
23 \( 1 + (-0.964 - 0.263i)T \)
29 \( 1 + (-0.617 + 0.786i)T \)
31 \( 1 + (0.288 + 0.957i)T \)
37 \( 1 + (-0.769 + 0.638i)T \)
41 \( 1 + (-0.237 - 0.971i)T \)
43 \( 1 + (0.977 + 0.211i)T \)
47 \( 1 + (-0.964 - 0.263i)T \)
53 \( 1 + (0.288 + 0.957i)T \)
59 \( 1 + (0.734 + 0.678i)T \)
61 \( 1 + (-0.530 - 0.847i)T \)
67 \( 1 + (-0.339 + 0.940i)T \)
71 \( 1 + (-0.931 - 0.364i)T \)
73 \( 1 + (0.861 + 0.507i)T \)
79 \( 1 + (0.802 + 0.596i)T \)
83 \( 1 + (0.185 + 0.982i)T \)
89 \( 1 + (0.994 + 0.106i)T \)
97 \( 1 + (0.388 + 0.921i)T \)
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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

−22.20169862460917265685441978471, −21.034135097362413680845180674468, −20.46132116011628538517412969839, −19.76975519310292234861570684742, −19.19669792772193702143133937961, −18.167051720729691441630975570787, −17.55665203743623670119841968268, −16.85842074817165961079148676467, −15.968694959984178240911243403722, −14.50663194135924298123323350046, −13.66547662146234385012064068474, −13.14980927214868365224607032498, −12.23547885971746897098495040838, −11.6366219608141990440891904371, −10.02880921123819866213934453057, −9.565746580744071579151265179398, −9.13160732125192903024412904968, −7.731215534726349619560566141595, −7.170548652804748690221576168351, −6.27621619526258561117154820939, −4.65520596160843743795779871986, −3.57034286419288725731220107818, −2.47757443435077327920936579736, −1.79412217760186146746848782224, −0.644698996086902090372528889617, 1.59474461536049726389637309509, 2.72189490448023450669277623049, 3.63669524449607355666824401602, 5.30584311556104932132670734928, 5.75255589953056349753474588064, 6.74066541191204936996111242513, 7.90779871359960508469066632315, 8.75599014483623876743308583334, 9.465188875411103838660905407705, 10.17680542567189461073153908223, 10.67770953546633737956957377530, 12.186180909968823952179110936253, 13.451902138613889274430022550198, 14.2587069620178246154257953454, 14.74627801253810346565796697934, 15.679402524514445976751507673107, 16.418554175775195551170262492327, 17.07609157149291656572740454093, 18.092452679174194047752653318511, 18.952637893753871284976379997693, 19.50186619842963845343590729838, 20.39940848715206520290543625049, 21.473774149194962380025252712691, 22.224089425512545510425731175916, 22.65414447475075649827659752758

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