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Properties

Label 2-975-39.5-c1-0-78
Degree $2$
Conductor $975$
Sign $-0.614 - 0.789i$
Analytic cond. $7.78541$
Root an. cond. $2.79023$
Motivic weight $1$
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)2-s + (1 − 1.41i)3-s − 0.999i·4-s + (−1.70 + 0.292i)6-s + (−1 − i)7-s + (−2.12 + 2.12i)8-s + (−1.00 − 2.82i)9-s + (−2.82 + 2.82i)11-s + (−1.41 − 0.999i)12-s + (2 − 3i)13-s + 1.41i·14-s + 1.00·16-s + (−1.29 + 2.70i)18-s + (1 − i)19-s + (−2.41 + 0.414i)21-s + 4.00·22-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (0.577 − 0.816i)3-s − 0.499i·4-s + (−0.696 + 0.119i)6-s + (−0.377 − 0.377i)7-s + (−0.750 + 0.750i)8-s + (−0.333 − 0.942i)9-s + (−0.852 + 0.852i)11-s + (−0.408 − 0.288i)12-s + (0.554 − 0.832i)13-s + 0.377i·14-s + 0.250·16-s + (−0.304 + 0.638i)18-s + (0.229 − 0.229i)19-s + (−0.526 + 0.0903i)21-s + 0.852·22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.614 - 0.789i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.614 - 0.789i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $-0.614 - 0.789i$
Analytic conductor: \(7.78541\)
Root analytic conductor: \(2.79023\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (551, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :1/2),\ -0.614 - 0.789i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.271828 + 0.555983i\)
\(L(\frac12)\) \(\approx\) \(0.271828 + 0.555983i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1 + 1.41i)T \)
5 \( 1 \)
13 \( 1 + (-2 + 3i)T \)
good2 \( 1 + (0.707 + 0.707i)T + 2iT^{2} \)
7 \( 1 + (1 + i)T + 7iT^{2} \)
11 \( 1 + (2.82 - 2.82i)T - 11iT^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + (-1 + i)T - 19iT^{2} \)
23 \( 1 + 8.48T + 23T^{2} \)
29 \( 1 + 2.82iT - 29T^{2} \)
31 \( 1 + (5 - 5i)T - 31iT^{2} \)
37 \( 1 + (1 + i)T + 37iT^{2} \)
41 \( 1 + (1.41 + 1.41i)T + 41iT^{2} \)
43 \( 1 - 6iT - 43T^{2} \)
47 \( 1 + (-2.82 + 2.82i)T - 47iT^{2} \)
53 \( 1 + 5.65iT - 53T^{2} \)
59 \( 1 + (2.82 - 2.82i)T - 59iT^{2} \)
61 \( 1 - 8T + 61T^{2} \)
67 \( 1 + (-5 + 5i)T - 67iT^{2} \)
71 \( 1 + (-2.82 - 2.82i)T + 71iT^{2} \)
73 \( 1 + (1 + i)T + 73iT^{2} \)
79 \( 1 + 10T + 79T^{2} \)
83 \( 1 + (-5.65 - 5.65i)T + 83iT^{2} \)
89 \( 1 + (-9.89 + 9.89i)T - 89iT^{2} \)
97 \( 1 + (7 - 7i)T - 97iT^{2} \)
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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

−9.673451268081745190128082809495, −8.599061143618181068014766861767, −7.961762332918055513978201182778, −7.07016635259605825682806147076, −6.11303450444885643996722076195, −5.26020901751273711743845626171, −3.73149944900423468378206603820, −2.61279009361964217256241870082, −1.69219593981253661912035641300, −0.29505375134976694326419730739, 2.35446875000867515281987056864, 3.42587723280838639040613155157, 4.10649225898848003879516446271, 5.51681358829249898529448746781, 6.28587054128424301410234678989, 7.51766203852099923779310812915, 8.168509582588643533159577507918, 8.846028324195117207948416607887, 9.456411782061042449518536235933, 10.28685822042339847621450064783

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