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Properties

Label 1-975-975.602-r0-0-0
Degree $1$
Conductor $975$
Sign $0.138 - 0.990i$
Analytic cond. $4.52788$
Root an. cond. $4.52788$
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  + (−0.207 − 0.978i)2-s + (−0.913 + 0.406i)4-s + (0.866 + 0.5i)7-s + (0.587 + 0.809i)8-s + (−0.978 + 0.207i)11-s + (0.309 − 0.951i)14-s + (0.669 − 0.743i)16-s + (−0.994 + 0.104i)17-s + (−0.104 − 0.994i)19-s + (0.406 + 0.913i)22-s + (−0.207 − 0.978i)23-s + (−0.994 − 0.104i)28-s + (−0.104 + 0.994i)29-s + (0.809 − 0.587i)31-s + (−0.866 − 0.5i)32-s + ⋯
L(s)  = 1  + (−0.207 − 0.978i)2-s + (−0.913 + 0.406i)4-s + (0.866 + 0.5i)7-s + (0.587 + 0.809i)8-s + (−0.978 + 0.207i)11-s + (0.309 − 0.951i)14-s + (0.669 − 0.743i)16-s + (−0.994 + 0.104i)17-s + (−0.104 − 0.994i)19-s + (0.406 + 0.913i)22-s + (−0.207 − 0.978i)23-s + (−0.994 − 0.104i)28-s + (−0.104 + 0.994i)29-s + (0.809 − 0.587i)31-s + (−0.866 − 0.5i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.138 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.138 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $0.138 - 0.990i$
Analytic conductor: \(4.52788\)
Root analytic conductor: \(4.52788\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (602, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 975,\ (0:\ ),\ 0.138 - 0.990i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8523363151 - 0.7417772526i\)
\(L(\frac12)\) \(\approx\) \(0.8523363151 - 0.7417772526i\)
\(L(1)\) \(\approx\) \(0.8204204497 - 0.3836254698i\)
\(L(1)\) \(\approx\) \(0.8204204497 - 0.3836254698i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.207 - 0.978i)T \)
7 \( 1 + (0.866 + 0.5i)T \)
11 \( 1 + (-0.978 + 0.207i)T \)
17 \( 1 + (-0.994 + 0.104i)T \)
19 \( 1 + (-0.104 - 0.994i)T \)
23 \( 1 + (-0.207 - 0.978i)T \)
29 \( 1 + (-0.104 + 0.994i)T \)
31 \( 1 + (0.809 - 0.587i)T \)
37 \( 1 + (-0.743 - 0.669i)T \)
41 \( 1 + (0.669 - 0.743i)T \)
43 \( 1 + (0.866 + 0.5i)T \)
47 \( 1 + (0.587 - 0.809i)T \)
53 \( 1 + (0.587 - 0.809i)T \)
59 \( 1 + (0.978 + 0.207i)T \)
61 \( 1 + (0.669 + 0.743i)T \)
67 \( 1 + (0.406 - 0.913i)T \)
71 \( 1 + (0.913 - 0.406i)T \)
73 \( 1 + (0.951 + 0.309i)T \)
79 \( 1 + (0.809 + 0.587i)T \)
83 \( 1 + (0.587 + 0.809i)T \)
89 \( 1 + (0.978 - 0.207i)T \)
97 \( 1 + (-0.406 - 0.913i)T \)
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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

−22.0076086627601813956982186193, −21.132170042620981916540499116036, −20.40841303168859363288035511165, −19.30287385720593295772965169266, −18.62797457010765198278027410472, −17.62290113989807960409248664655, −17.42774246489333637244719150304, −16.291599424911117253033035543, −15.6237789162444180240002939113, −14.934373563432483156947544489675, −13.89593806091430317329353461985, −13.57524868256273917020260734476, −12.49621561600560485242481058769, −11.303479570308470173015870721303, −10.48676713625148223770225463954, −9.72660130879472627555288837150, −8.601073504878280080165827717217, −7.96400938656188226360947505199, −7.326305577394359873317403471774, −6.26736120862812526330557691765, −5.393857505164389044424644018714, −4.60408345968157385397916533794, −3.73709925222188176387553090200, −2.19317799951911391876872433167, −0.93921162108448751342743127088, 0.68310543511567874971423034456, 2.23292229723128239403259761584, 2.44498727370275774070668679250, 3.91273710365979275879240737480, 4.80101897428372947392279955052, 5.43356586053583167282306679416, 6.89492416153874266838513949036, 7.97309570622075860763209987007, 8.6473728618340208122137772001, 9.361828388625733977118920643035, 10.59584788242934325702011859100, 10.91557375833281950318832689322, 11.88257353046145868770850584921, 12.65343862819498196691277281884, 13.38570795507158188222740661289, 14.25764495516413622250137479045, 15.15418977743027060232220088577, 15.99714350518308401247665803198, 17.190206033714556042301151102257, 17.9082662856013624521806378838, 18.3094013922823375095221771570, 19.262829187834036359180101433191, 20.04217188765943466357891297314, 20.86301203533736770156015265328, 21.30248536532264762311067889172

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