Properties

Label 1-968-968.171-r0-0-0
Degree $1$
Conductor $968$
Sign $-0.436 + 0.899i$
Analytic cond. $4.49537$
Root an. cond. $4.49537$
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  + (0.309 + 0.951i)3-s + (−0.0855 − 0.996i)5-s + (0.993 − 0.113i)7-s + (−0.809 + 0.587i)9-s + (−0.362 + 0.931i)13-s + (0.921 − 0.389i)15-s + (−0.696 + 0.717i)17-s + (−0.897 + 0.441i)19-s + (0.415 + 0.909i)21-s + (−0.415 + 0.909i)23-s + (−0.985 + 0.170i)25-s + (−0.809 − 0.587i)27-s + (−0.985 − 0.170i)29-s + (0.998 − 0.0570i)31-s + (−0.198 − 0.980i)35-s + ⋯
L(s)  = 1  + (0.309 + 0.951i)3-s + (−0.0855 − 0.996i)5-s + (0.993 − 0.113i)7-s + (−0.809 + 0.587i)9-s + (−0.362 + 0.931i)13-s + (0.921 − 0.389i)15-s + (−0.696 + 0.717i)17-s + (−0.897 + 0.441i)19-s + (0.415 + 0.909i)21-s + (−0.415 + 0.909i)23-s + (−0.985 + 0.170i)25-s + (−0.809 − 0.587i)27-s + (−0.985 − 0.170i)29-s + (0.998 − 0.0570i)31-s + (−0.198 − 0.980i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.436 + 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.436 + 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(968\)    =    \(2^{3} \cdot 11^{2}\)
Sign: $-0.436 + 0.899i$
Analytic conductor: \(4.49537\)
Root analytic conductor: \(4.49537\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{968} (171, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 968,\ (0:\ ),\ -0.436 + 0.899i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6643974065 + 1.060873060i\)
\(L(\frac12)\) \(\approx\) \(0.6643974065 + 1.060873060i\)
\(L(1)\) \(\approx\) \(1.006850661 + 0.3810419054i\)
\(L(1)\) \(\approx\) \(1.006850661 + 0.3810419054i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 \)
good3 \( 1 + (0.309 + 0.951i)T \)
5 \( 1 + (-0.0855 - 0.996i)T \)
7 \( 1 + (0.993 - 0.113i)T \)
13 \( 1 + (-0.362 + 0.931i)T \)
17 \( 1 + (-0.696 + 0.717i)T \)
19 \( 1 + (-0.897 + 0.441i)T \)
23 \( 1 + (-0.415 + 0.909i)T \)
29 \( 1 + (-0.985 - 0.170i)T \)
31 \( 1 + (0.998 - 0.0570i)T \)
37 \( 1 + (-0.774 + 0.633i)T \)
41 \( 1 + (0.564 + 0.825i)T \)
43 \( 1 + (0.654 - 0.755i)T \)
47 \( 1 + (0.0285 + 0.999i)T \)
53 \( 1 + (0.870 + 0.491i)T \)
59 \( 1 + (-0.564 + 0.825i)T \)
61 \( 1 + (0.610 + 0.791i)T \)
67 \( 1 + (-0.959 + 0.281i)T \)
71 \( 1 + (0.466 - 0.884i)T \)
73 \( 1 + (0.736 + 0.676i)T \)
79 \( 1 + (0.974 + 0.226i)T \)
83 \( 1 + (-0.198 + 0.980i)T \)
89 \( 1 + (-0.142 + 0.989i)T \)
97 \( 1 + (0.0855 - 0.996i)T \)
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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

−21.464148871277652275682653029671, −20.57540018786605760983012426829, −19.84710835177253714265282973946, −19.04267510261594995795252058830, −18.3063474309604490091712464883, −17.74471563315384261627135332401, −17.18380899789473352980221464027, −15.68861550193215829154502831024, −14.925950755230598386505765050552, −14.377912908970146404453603756, −13.62757043812647815256657690117, −12.714603256438664775551100201488, −11.85828024276292026321572607900, −11.08266404653898671232933638395, −10.41258989881165571642486957777, −9.10038256347997974462880608688, −8.24646980603356590061473199935, −7.53864456982211453700731752243, −6.82297214409058551882960753134, −5.95607962253730048778241241805, −4.88585924339563080665577732830, −3.65706562950274363038984492315, −2.48329037856312677464928648526, −2.10356860729187215254779755409, −0.491453908694761736765436726035, 1.50039154601696604885432608897, 2.33977650227553031716334991229, 3.99402163544694930176661960622, 4.29763234898583542645869012530, 5.18020584849447516913722108993, 6.07824966102743697517075432878, 7.55901817706321032110369201968, 8.35529409712722302546629469432, 8.9453345718405956453547027912, 9.75732282417281744860790730187, 10.73041981003540137631710055543, 11.499905214408370437662333278282, 12.26223233557574000741408246279, 13.4400015264943519588495182906, 14.073959064839657927638753501085, 15.04511532643482317170251739627, 15.53260092503773886688739478444, 16.65387877009493773650940318015, 17.01828073492708561607751126562, 17.80459044526451890255852198654, 19.20875677361017180490668256762, 19.704555471519155803413250569, 20.69748955805233976580381578117, 21.10555195533225277771803580438, 21.693200325668108801455247260176

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