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Properties

Label 1-967-967.938-r0-0-0
Degree $1$
Conductor $967$
Sign $0.319 + 0.947i$
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  + (−0.696 − 0.717i)2-s + (0.999 − 0.0390i)3-s + (−0.0292 + 0.999i)4-s + (0.165 + 0.986i)5-s + (−0.724 − 0.689i)6-s + (0.993 − 0.116i)7-s + (0.737 − 0.675i)8-s + (0.996 − 0.0779i)9-s + (0.592 − 0.805i)10-s + (−0.932 − 0.362i)11-s + (0.00975 + 0.999i)12-s + (−0.932 + 0.362i)13-s + (−0.775 − 0.631i)14-s + (0.203 + 0.979i)15-s + (−0.998 − 0.0585i)16-s + (−0.608 + 0.793i)17-s + ⋯
L(s)  = 1  + (−0.696 − 0.717i)2-s + (0.999 − 0.0390i)3-s + (−0.0292 + 0.999i)4-s + (0.165 + 0.986i)5-s + (−0.724 − 0.689i)6-s + (0.993 − 0.116i)7-s + (0.737 − 0.675i)8-s + (0.996 − 0.0779i)9-s + (0.592 − 0.805i)10-s + (−0.932 − 0.362i)11-s + (0.00975 + 0.999i)12-s + (−0.932 + 0.362i)13-s + (−0.775 − 0.631i)14-s + (0.203 + 0.979i)15-s + (−0.998 − 0.0585i)16-s + (−0.608 + 0.793i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.319 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.319 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $0.319 + 0.947i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (938, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ 0.319 + 0.947i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.008699396 + 0.7247485219i\)
\(L(\frac12)\) \(\approx\) \(1.008699396 + 0.7247485219i\)
\(L(1)\) \(\approx\) \(1.014679584 + 0.07349043944i\)
\(L(1)\) \(\approx\) \(1.014679584 + 0.07349043944i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (-0.696 - 0.717i)T \)
3 \( 1 + (0.999 - 0.0390i)T \)
5 \( 1 + (0.165 + 0.986i)T \)
7 \( 1 + (0.993 - 0.116i)T \)
11 \( 1 + (-0.932 - 0.362i)T \)
13 \( 1 + (-0.932 + 0.362i)T \)
17 \( 1 + (-0.608 + 0.793i)T \)
19 \( 1 + (-0.184 + 0.982i)T \)
23 \( 1 + (-0.799 + 0.600i)T \)
29 \( 1 + (-0.297 + 0.954i)T \)
31 \( 1 + (-0.371 + 0.928i)T \)
37 \( 1 + (0.560 + 0.828i)T \)
41 \( 1 + (-0.990 + 0.136i)T \)
43 \( 1 + (0.241 - 0.970i)T \)
47 \( 1 + (-0.999 - 0.0195i)T \)
53 \( 1 + (0.460 - 0.887i)T \)
59 \( 1 + (0.763 - 0.646i)T \)
61 \( 1 + (-0.990 + 0.136i)T \)
67 \( 1 + (-0.371 - 0.928i)T \)
71 \( 1 + (-0.696 + 0.717i)T \)
73 \( 1 + (0.460 + 0.887i)T \)
79 \( 1 + (0.981 + 0.193i)T \)
83 \( 1 + (0.951 + 0.307i)T \)
89 \( 1 + (-0.371 - 0.928i)T \)
97 \( 1 + (0.623 + 0.781i)T \)
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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

−21.28093353703229566168902102390, −20.53160684559301648506905291822, −20.07101396757105831552896732230, −19.316720879906357430748403497824, −18.112574004709442799505241877356, −17.89089838722560849417954165018, −16.839660870942545020488302091061, −15.99734360608626484368201575455, −15.226016407883493301048525484726, −14.75337454892527144743488481545, −13.67047618587286208499163233221, −13.17388539680392351842327115098, −12.00081911858608597762278201032, −10.85250774770363797269072919442, −9.85948122281437323499445058257, −9.27768077060986127024750656800, −8.459248425179803634963638649793, −7.7985850492022294301374725929, −7.28227635901893118712936249728, −5.84439391067402529463876073418, −4.72296913346236178450024476598, −4.56131092761385180920912640485, −2.43781481067875622783714779541, −1.98557706730902653895527979987, −0.54314423906472343178083256063, 1.72810027475404862557195574277, 2.09975892855866965111487070063, 3.17340836404400896421683859326, 3.89810187269247178219623803867, 5.05527827509798209743281467756, 6.65294830180651261503458202719, 7.54488408520982279359472528017, 8.073192150832603022872270424182, 8.83980834463006797289470373859, 10.01887723862832379889869148449, 10.38058162228148244740769143910, 11.257805302440006897237957939526, 12.197972105049359652756211651751, 13.16613086824681915449356390086, 13.96819624786346658503183909539, 14.6616941310200809159113284098, 15.435853182104617764919561267421, 16.50722459524403429273373727819, 17.54795934164247851628185236959, 18.24904055605673756402712040972, 18.7216051225411895565103995425, 19.57986442019244081279550526029, 20.191346348255697596117897700589, 21.13714311670386242651329552967, 21.59229328064301640108442794027

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