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Properties

Label 1-960-960.149-r1-0-0
Degree $1$
Conductor $960$
Sign $-0.0980 + 0.995i$
Analytic cond. $103.166$
Root an. cond. $103.166$
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  + (−0.707 − 0.707i)7-s + (−0.923 − 0.382i)11-s + (−0.382 − 0.923i)13-s − i·17-s + (−0.382 − 0.923i)19-s + (−0.707 + 0.707i)23-s + (−0.923 + 0.382i)29-s − 31-s + (0.382 − 0.923i)37-s + (0.707 − 0.707i)41-s + (0.923 + 0.382i)43-s − i·47-s + i·49-s + (0.923 + 0.382i)53-s + (−0.382 + 0.923i)59-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)7-s + (−0.923 − 0.382i)11-s + (−0.382 − 0.923i)13-s − i·17-s + (−0.382 − 0.923i)19-s + (−0.707 + 0.707i)23-s + (−0.923 + 0.382i)29-s − 31-s + (0.382 − 0.923i)37-s + (0.707 − 0.707i)41-s + (0.923 + 0.382i)43-s − i·47-s + i·49-s + (0.923 + 0.382i)53-s + (−0.382 + 0.923i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0980 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0980 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $-0.0980 + 0.995i$
Analytic conductor: \(103.166\)
Root analytic conductor: \(103.166\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{960} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 960,\ (1:\ ),\ -0.0980 + 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02590577650 + 0.02858261975i\)
\(L(\frac12)\) \(\approx\) \(0.02590577650 + 0.02858261975i\)
\(L(1)\) \(\approx\) \(0.7166411420 - 0.1971630242i\)
\(L(1)\) \(\approx\) \(0.7166411420 - 0.1971630242i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (-0.707 - 0.707i)T \)
11 \( 1 + (-0.923 - 0.382i)T \)
13 \( 1 + (-0.382 - 0.923i)T \)
17 \( 1 - iT \)
19 \( 1 + (-0.382 - 0.923i)T \)
23 \( 1 + (-0.707 + 0.707i)T \)
29 \( 1 + (-0.923 + 0.382i)T \)
31 \( 1 - T \)
37 \( 1 + (0.382 - 0.923i)T \)
41 \( 1 + (0.707 - 0.707i)T \)
43 \( 1 + (0.923 + 0.382i)T \)
47 \( 1 - iT \)
53 \( 1 + (0.923 + 0.382i)T \)
59 \( 1 + (-0.382 + 0.923i)T \)
61 \( 1 + (-0.923 + 0.382i)T \)
67 \( 1 + (0.923 - 0.382i)T \)
71 \( 1 + (-0.707 - 0.707i)T \)
73 \( 1 + (-0.707 + 0.707i)T \)
79 \( 1 + iT \)
83 \( 1 + (-0.382 - 0.923i)T \)
89 \( 1 + (0.707 + 0.707i)T \)
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

−21.43171932609442704427936477920, −20.63684401644699843741223288739, −19.74123287844023692248277045787, −18.79700531270012329770135933286, −18.5373718156732452553684221944, −17.375887278074413008193528473083, −16.52543251236769277142581625780, −15.905577086193338636528337370464, −14.95816803432272622298912775780, −14.38703231796498427744080334038, −13.12871264110494473233607005787, −12.63204354750149239897851342689, −11.87055034382066077966037110008, −10.78380250079415858403795175053, −9.97174674345307903595237579153, −9.24053497767353320896168914333, −8.27860137865205904515190315776, −7.45924023919548246715642995877, −6.31542054543623052941207211602, −5.76670312748212551589792240881, −4.587226936832344518521841953494, −3.6902900695761128297500497267, −2.49928280401436692489195515697, −1.79312453441752566872084345835, −0.01114976030804635271047849263, 0.69963458257967068332028789585, 2.32798489827177614708864504692, 3.149823549660015910562385769303, 4.10591088818822680197788264416, 5.26421827544306636932709589560, 5.93378046354573909022881788244, 7.35141821987915836058023954841, 7.48259663386635322167895698208, 8.871204576014224662477527331195, 9.643015390740933743274532515776, 10.55060207750226950020376581001, 11.10695591642189133368712008613, 12.316285365921652730122152081, 13.12563507860693469698806374736, 13.60141556986756421115378192141, 14.6349488083046924239256629084, 15.63205492976793005672192696766, 16.14954114202282741346647559013, 17.02393286897003972712683723867, 17.915523771167833420624494548206, 18.54253639865537868062910119453, 19.69368639532925152614443634629, 20.01404315138393070833172817255, 20.9625702773102534000983152954, 21.82868496028036918791643513969

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