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Properties

Label 1-709-709.180-r0-0-0
Degree $1$
Conductor $709$
Sign $0.993 + 0.115i$
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  + (0.937 − 0.347i)2-s + (−0.202 + 0.979i)3-s + (0.758 − 0.651i)4-s + (0.355 − 0.934i)5-s + (0.150 + 0.988i)6-s + (0.924 + 0.380i)7-s + (0.484 − 0.874i)8-s + (−0.917 − 0.396i)9-s + (0.00887 − 0.999i)10-s + (0.603 + 0.797i)11-s + (0.484 + 0.874i)12-s + (−0.132 + 0.991i)13-s + (0.999 + 0.0354i)14-s + (0.842 + 0.537i)15-s + (0.150 − 0.988i)16-s + (−0.992 − 0.123i)17-s + ⋯
L(s)  = 1  + (0.937 − 0.347i)2-s + (−0.202 + 0.979i)3-s + (0.758 − 0.651i)4-s + (0.355 − 0.934i)5-s + (0.150 + 0.988i)6-s + (0.924 + 0.380i)7-s + (0.484 − 0.874i)8-s + (−0.917 − 0.396i)9-s + (0.00887 − 0.999i)10-s + (0.603 + 0.797i)11-s + (0.484 + 0.874i)12-s + (−0.132 + 0.991i)13-s + (0.999 + 0.0354i)14-s + (0.842 + 0.537i)15-s + (0.150 − 0.988i)16-s + (−0.992 − 0.123i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.115i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.115i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(709\)
Sign: $0.993 + 0.115i$
Analytic conductor: \(3.29258\)
Root analytic conductor: \(3.29258\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{709} (180, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 709,\ (0:\ ),\ 0.993 + 0.115i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.864833922 + 0.1664213399i\)
\(L(\frac12)\) \(\approx\) \(2.864833922 + 0.1664213399i\)
\(L(1)\) \(\approx\) \(1.990909380 + 0.01551139544i\)
\(L(1)\) \(\approx\) \(1.990909380 + 0.01551139544i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad709 \( 1 \)
good2 \( 1 + (0.937 - 0.347i)T \)
3 \( 1 + (-0.202 + 0.979i)T \)
5 \( 1 + (0.355 - 0.934i)T \)
7 \( 1 + (0.924 + 0.380i)T \)
11 \( 1 + (0.603 + 0.797i)T \)
13 \( 1 + (-0.132 + 0.991i)T \)
17 \( 1 + (-0.992 - 0.123i)T \)
19 \( 1 + (0.515 + 0.857i)T \)
23 \( 1 + (0.684 + 0.728i)T \)
29 \( 1 + (0.545 + 0.838i)T \)
31 \( 1 + (0.0443 - 0.999i)T \)
37 \( 1 + (0.924 + 0.380i)T \)
41 \( 1 + (-0.589 - 0.807i)T \)
43 \( 1 + (-0.903 - 0.429i)T \)
47 \( 1 + (0.288 - 0.957i)T \)
53 \( 1 + (-0.887 + 0.461i)T \)
59 \( 1 + (0.484 - 0.874i)T \)
61 \( 1 + (0.969 - 0.245i)T \)
67 \( 1 + (0.322 - 0.946i)T \)
71 \( 1 + (0.00887 + 0.999i)T \)
73 \( 1 + (-0.833 - 0.552i)T \)
79 \( 1 + (-0.722 + 0.691i)T \)
83 \( 1 + (-0.964 - 0.263i)T \)
89 \( 1 + (-0.954 + 0.297i)T \)
97 \( 1 + (0.984 - 0.176i)T \)
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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

−22.59756067414785217284467169734, −22.07234293826457526637282149706, −21.2245436141188807944578259447, −20.103759010892523904015149247750, −19.51879091774646969152430182648, −18.29525456125877800122274856576, −17.58200220836865749356058384803, −17.13828687087362785851368182350, −15.89610116944307629452928435220, −14.79123333004761006148443252346, −14.3429098203692056174570835826, −13.476850586944778094304243074704, −13.00276036947502165539546675852, −11.637538107117185195985330767565, −11.2785810043376338111764194431, −10.48130084627022358280328532722, −8.6646224494322828309788467579, −7.872658300770862991459247689787, −6.98027425966887070410733568091, −6.41043461192040850979402731096, −5.51090192813609471857464233681, −4.53506423471605768713253153968, −3.12004203848218254570707934280, −2.48571410484446670422004649913, −1.2216693019028450605709822461, 1.42450557644785707943778754104, 2.24659358709447491219823874365, 3.746441047148283908129074320494, 4.55905704272456576379625459695, 5.02914051163773147686340919211, 5.90211522036456941583603510940, 7.01391206528066761151471957210, 8.49957513376378955880616848843, 9.375479113405167372549602995603, 10.016103768972874274995807208539, 11.29239019776681981325279701012, 11.712061636139321020755727222857, 12.51478095495955699000667124138, 13.67930279374751377135114055777, 14.38937504617399797210768307619, 15.14620776885650650324446149694, 15.88389613402193427484421286610, 16.8201910462108141386053818282, 17.42105734470888484885892643213, 18.640676408245401976497099869957, 20.02729460763332045384948160453, 20.34025164619678459836036381472, 21.136087816481863385291358106350, 21.75709902113148161270037136962, 22.32345621703634156082805133288

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