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Properties

Label 2-975-975.116-c0-0-2
Degree $2$
Conductor $975$
Sign $-0.187 - 0.982i$
Analytic cond. $0.486588$
Root an. cond. $0.697558$
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  + (1.30 + 0.951i)2-s + (0.309 + 0.951i)3-s + (0.500 + 1.53i)4-s + (0.309 − 0.951i)5-s + (−0.499 + 1.53i)6-s + (−0.309 + 0.951i)8-s + (−0.809 + 0.587i)9-s + (1.30 − 0.951i)10-s + (−0.5 − 0.363i)11-s + (−1.30 + 0.951i)12-s + (−0.809 + 0.587i)13-s + 0.999·15-s − 1.61·18-s + 1.61·20-s + (−0.309 − 0.951i)22-s + ⋯
L(s)  = 1  + (1.30 + 0.951i)2-s + (0.309 + 0.951i)3-s + (0.500 + 1.53i)4-s + (0.309 − 0.951i)5-s + (−0.499 + 1.53i)6-s + (−0.309 + 0.951i)8-s + (−0.809 + 0.587i)9-s + (1.30 − 0.951i)10-s + (−0.5 − 0.363i)11-s + (−1.30 + 0.951i)12-s + (−0.809 + 0.587i)13-s + 0.999·15-s − 1.61·18-s + 1.61·20-s + (−0.309 − 0.951i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.187 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.187 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $-0.187 - 0.982i$
Analytic conductor: \(0.486588\)
Root analytic conductor: \(0.697558\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (116, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :0),\ -0.187 - 0.982i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.112740962\)
\(L(\frac12)\) \(\approx\) \(2.112740962\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.309 - 0.951i)T \)
5 \( 1 + (-0.309 + 0.951i)T \)
13 \( 1 + (0.809 - 0.587i)T \)
good2 \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
17 \( 1 + (0.809 + 0.587i)T^{2} \)
19 \( 1 + (0.809 + 0.587i)T^{2} \)
23 \( 1 + (-0.309 - 0.951i)T^{2} \)
29 \( 1 + (0.809 - 0.587i)T^{2} \)
31 \( 1 + (0.809 + 0.587i)T^{2} \)
37 \( 1 + (-0.309 + 0.951i)T^{2} \)
41 \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \)
43 \( 1 + 1.61T + T^{2} \)
47 \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \)
53 \( 1 + (0.809 - 0.587i)T^{2} \)
59 \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \)
61 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
67 \( 1 + (0.809 + 0.587i)T^{2} \)
71 \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \)
73 \( 1 + (-0.309 - 0.951i)T^{2} \)
79 \( 1 + (-0.618 - 1.90i)T + (-0.809 + 0.587i)T^{2} \)
83 \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \)
89 \( 1 + (1.61 + 1.17i)T + (0.309 + 0.951i)T^{2} \)
97 \( 1 + (0.809 - 0.587i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.28331181934518982527398823568, −9.535236884081642791911470931074, −8.637557535978894234793316318760, −7.905387872683363810458876829502, −6.87824042731228196679747510471, −5.72020773079055556573119569989, −5.20573485845500705884573136699, −4.48069334615586022715052306901, −3.69298868551294551782525900846, −2.44276003100706693219236230512, 1.76161507684809680897685615600, 2.71188926648980979418460893283, 3.21380728334865183838558036061, 4.57737369991491520506250075104, 5.62592859934174111986876808151, 6.33610630923926696797248345943, 7.33464023771164139048389625151, 8.035724127590827111933347834814, 9.469179241575742603919368352758, 10.28163098611060544721536188176

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