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Properties

Label 1-967-967.71-r0-0-0
Degree $1$
Conductor $967$
Sign $-0.968 + 0.249i$
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  + (−0.977 − 0.212i)2-s + (0.0487 + 0.998i)3-s + (0.909 + 0.416i)4-s + (−0.184 + 0.982i)5-s + (0.165 − 0.986i)6-s + (−0.145 − 0.989i)7-s + (−0.799 − 0.600i)8-s + (−0.995 + 0.0974i)9-s + (0.389 − 0.921i)10-s + (0.316 + 0.948i)11-s + (−0.371 + 0.928i)12-s + (0.316 − 0.948i)13-s + (−0.0682 + 0.997i)14-s + (−0.990 − 0.136i)15-s + (0.653 + 0.756i)16-s + (0.353 + 0.935i)17-s + ⋯
L(s)  = 1  + (−0.977 − 0.212i)2-s + (0.0487 + 0.998i)3-s + (0.909 + 0.416i)4-s + (−0.184 + 0.982i)5-s + (0.165 − 0.986i)6-s + (−0.145 − 0.989i)7-s + (−0.799 − 0.600i)8-s + (−0.995 + 0.0974i)9-s + (0.389 − 0.921i)10-s + (0.316 + 0.948i)11-s + (−0.371 + 0.928i)12-s + (0.316 − 0.948i)13-s + (−0.0682 + 0.997i)14-s + (−0.990 − 0.136i)15-s + (0.653 + 0.756i)16-s + (0.353 + 0.935i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.968 + 0.249i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.968 + 0.249i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $-0.968 + 0.249i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ -0.968 + 0.249i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.07348159829 + 0.5806751634i\)
\(L(\frac12)\) \(\approx\) \(0.07348159829 + 0.5806751634i\)
\(L(1)\) \(\approx\) \(0.5486833821 + 0.2960856785i\)
\(L(1)\) \(\approx\) \(0.5486833821 + 0.2960856785i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (-0.977 - 0.212i)T \)
3 \( 1 + (0.0487 + 0.998i)T \)
5 \( 1 + (-0.184 + 0.982i)T \)
7 \( 1 + (-0.145 - 0.989i)T \)
11 \( 1 + (0.316 + 0.948i)T \)
13 \( 1 + (0.316 - 0.948i)T \)
17 \( 1 + (0.353 + 0.935i)T \)
19 \( 1 + (0.811 + 0.584i)T \)
23 \( 1 + (-0.999 + 0.0195i)T \)
29 \( 1 + (-0.696 + 0.717i)T \)
31 \( 1 + (0.763 - 0.646i)T \)
37 \( 1 + (0.938 - 0.344i)T \)
41 \( 1 + (-0.576 + 0.816i)T \)
43 \( 1 + (0.0876 + 0.996i)T \)
47 \( 1 + (-0.724 + 0.689i)T \)
53 \( 1 + (0.203 - 0.979i)T \)
59 \( 1 + (-0.638 + 0.769i)T \)
61 \( 1 + (-0.576 + 0.816i)T \)
67 \( 1 + (0.763 + 0.646i)T \)
71 \( 1 + (-0.977 + 0.212i)T \)
73 \( 1 + (0.203 + 0.979i)T \)
79 \( 1 + (0.241 - 0.970i)T \)
83 \( 1 + (0.924 + 0.380i)T \)
89 \( 1 + (0.763 + 0.646i)T \)
97 \( 1 + (-0.900 + 0.433i)T \)
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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

−21.14895457922133662437654537440, −20.29181003074453861981116480845, −19.64439964230531050423976974107, −18.786793708856246388285212315730, −18.500626531484024721641059059028, −17.523196680561453405638842675660, −16.68443090819558115047495875103, −16.11498311632779484944758757231, −15.3450650885727656908280717378, −14.05907395889853843512438227199, −13.53298335690632838625585940910, −12.11594345304576019909575268216, −11.903622642726906918485702072868, −11.18376830628785057021466138604, −9.55618546779871452755828692504, −9.0641846676850079197090801429, −8.37131205956684784466474861321, −7.6807269098170688367742887165, −6.60424733272315263066670395842, −5.89862664470266383277716556445, −5.10753706520732180367992328522, −3.359550323974935127885151419758, −2.281474165415431073176137600774, −1.401427851679834432554455081971, −0.36558334011533195240234576245, 1.378308280399678498311486072988, 2.73446362535696163331804266812, 3.55126780448161512268683170941, 4.18266111136665580861432353834, 5.78298170300586161336819573374, 6.59432191158956575553416927720, 7.74420667527808076443316857441, 8.060427358692289409749480081439, 9.59573430014270999003732798432, 9.9723003756713501813601465210, 10.59873298424944467661162210228, 11.2803104509539950261186376267, 12.17583905284421813510156783263, 13.368133590178509568383746666690, 14.70900688717647750478572384495, 14.89406556720300256554132226497, 16.01772088789038758754781297475, 16.54184860460680078299040983581, 17.5846939228191776249046001567, 17.926535758761073159313335894410, 19.067636398117676800769300766996, 20.03063557164167467429443705069, 20.18929369422614497899090876485, 21.1457858266251576866475435906, 22.073906452487939670703234405603

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