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Properties

Label 1-967-967.196-r0-0-0
Degree $1$
Conductor $967$
Sign $0.580 + 0.814i$
Analytic cond. $4.49072$
Root an. cond. $4.49072$
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  + (−0.990 + 0.136i)2-s + (−0.775 + 0.631i)3-s + (0.962 − 0.269i)4-s + (0.854 − 0.519i)5-s + (0.682 − 0.730i)6-s + (0.460 + 0.887i)7-s + (−0.917 + 0.398i)8-s + (0.203 − 0.979i)9-s + (−0.775 + 0.631i)10-s + (0.203 − 0.979i)11-s + (−0.576 + 0.816i)12-s + (0.203 + 0.979i)13-s + (−0.576 − 0.816i)14-s + (−0.334 + 0.942i)15-s + (0.854 − 0.519i)16-s + (−0.334 − 0.942i)17-s + ⋯
L(s)  = 1  + (−0.990 + 0.136i)2-s + (−0.775 + 0.631i)3-s + (0.962 − 0.269i)4-s + (0.854 − 0.519i)5-s + (0.682 − 0.730i)6-s + (0.460 + 0.887i)7-s + (−0.917 + 0.398i)8-s + (0.203 − 0.979i)9-s + (−0.775 + 0.631i)10-s + (0.203 − 0.979i)11-s + (−0.576 + 0.816i)12-s + (0.203 + 0.979i)13-s + (−0.576 − 0.816i)14-s + (−0.334 + 0.942i)15-s + (0.854 − 0.519i)16-s + (−0.334 − 0.942i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.580 + 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.580 + 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $0.580 + 0.814i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (196, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ 0.580 + 0.814i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7934545217 + 0.4087947829i\)
\(L(\frac12)\) \(\approx\) \(0.7934545217 + 0.4087947829i\)
\(L(1)\) \(\approx\) \(0.6843676742 + 0.1730330744i\)
\(L(1)\) \(\approx\) \(0.6843676742 + 0.1730330744i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (-0.990 + 0.136i)T \)
3 \( 1 + (-0.775 + 0.631i)T \)
5 \( 1 + (0.854 - 0.519i)T \)
7 \( 1 + (0.460 + 0.887i)T \)
11 \( 1 + (0.203 - 0.979i)T \)
13 \( 1 + (0.203 + 0.979i)T \)
17 \( 1 + (-0.334 - 0.942i)T \)
19 \( 1 + (-0.775 + 0.631i)T \)
23 \( 1 + (0.962 - 0.269i)T \)
29 \( 1 + (0.203 + 0.979i)T \)
31 \( 1 + (-0.917 + 0.398i)T \)
37 \( 1 + (0.203 + 0.979i)T \)
41 \( 1 + (0.682 - 0.730i)T \)
43 \( 1 + (-0.334 + 0.942i)T \)
47 \( 1 + (-0.334 + 0.942i)T \)
53 \( 1 + (0.962 - 0.269i)T \)
59 \( 1 + (0.962 + 0.269i)T \)
61 \( 1 + (0.682 - 0.730i)T \)
67 \( 1 + (-0.917 - 0.398i)T \)
71 \( 1 + (-0.990 - 0.136i)T \)
73 \( 1 + (0.962 + 0.269i)T \)
79 \( 1 + (0.962 + 0.269i)T \)
83 \( 1 + (0.682 - 0.730i)T \)
89 \( 1 + (-0.917 - 0.398i)T \)
97 \( 1 + T \)
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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

−21.50644565219017203514577014215, −20.78852828240673455586176479449, −19.8056093217890035983952470677, −19.243583927317594065276224955541, −18.0772778213152552804784528294, −17.77332711613979808964031524677, −17.19585211338583592095528762579, −16.65828989021201844671832719294, −15.27596701379323081833149016244, −14.71173371580541268989035070223, −13.2230406086000877308107424753, −12.97651223311735202974728154737, −11.756672353384440945682736339522, −10.78606959963476109937813153584, −10.58526019086566587043262359404, −9.68843855505278927883906788664, −8.52683248739854891207651611536, −7.51247823073207546263755990981, −6.9932180377174859454893324308, −6.21252055467477061517783438276, −5.28978189747805283370294201668, −3.95119746591432117016216618078, −2.43068234605662523833812915450, −1.779649390359084979354075579068, −0.745875332722452916686597197834, 0.97980171784568902863206338903, 1.93588266398436451824068414874, 3.11338832202078716131744885654, 4.658602615221059185322912091605, 5.46014851296541047955573214739, 6.18085589575667332113092821230, 6.87628967825568156700381281617, 8.45005508540399196021530416007, 8.998684452824629071138943317185, 9.494528965406767956469177117593, 10.59706230798645169913175540513, 11.23962880323966670471785329915, 11.92561559557256809284112687775, 12.83591170873451692164077492880, 14.232543482158085357427462948469, 14.86270599064988550475616375991, 16.03902940751010225246518457252, 16.41812094835002176874938112780, 17.05553975350960492240206427813, 17.9657795740269296677845909066, 18.431833950528502838671240066157, 19.279369916731831102531859684553, 20.53514281962080915920256628422, 21.16484879686659720451910387988, 21.52886952057631505222471493379

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