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Properties

Label 2-975-195.77-c1-0-56
Degree $2$
Conductor $975$
Sign $0.525 - 0.850i$
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  + (1.63 + 1.63i)2-s + (1.22 − 1.22i)3-s + 3.32i·4-s + 3.99·6-s + (−2.16 + 2.16i)8-s − 2.99i·9-s + 5.62·11-s + (4.07 + 4.07i)12-s + (−2.54 + 2.54i)13-s − 0.404·16-s + (4.89 − 4.89i)18-s + (9.17 + 9.17i)22-s + 5.29i·24-s − 8.31·26-s + (−3.67 − 3.67i)27-s + ⋯
L(s)  = 1  + (1.15 + 1.15i)2-s + (0.707 − 0.707i)3-s + 1.66i·4-s + 1.63·6-s + (−0.764 + 0.764i)8-s − 0.999i·9-s + 1.69·11-s + (1.17 + 1.17i)12-s + (−0.707 + 0.707i)13-s − 0.101·16-s + (1.15 − 1.15i)18-s + (1.95 + 1.95i)22-s + 1.08i·24-s − 1.63·26-s + (−0.707 − 0.707i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $0.525 - 0.850i$
Analytic conductor: \(7.78541\)
Root analytic conductor: \(2.79023\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (857, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :1/2),\ 0.525 - 0.850i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.39328 + 1.89188i\)
\(L(\frac12)\) \(\approx\) \(3.39328 + 1.89188i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.22 + 1.22i)T \)
5 \( 1 \)
13 \( 1 + (2.54 - 2.54i)T \)
good2 \( 1 + (-1.63 - 1.63i)T + 2iT^{2} \)
7 \( 1 - 7iT^{2} \)
11 \( 1 - 5.62T + 11T^{2} \)
17 \( 1 - 17iT^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + 23iT^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 - 37iT^{2} \)
41 \( 1 - 6.37T + 41T^{2} \)
43 \( 1 + (8.24 - 8.24i)T - 43iT^{2} \)
47 \( 1 + (9.08 + 9.08i)T + 47iT^{2} \)
53 \( 1 + 53iT^{2} \)
59 \( 1 + 14.3iT - 59T^{2} \)
61 \( 1 + 14.8T + 61T^{2} \)
67 \( 1 - 67iT^{2} \)
71 \( 1 - 1.62T + 71T^{2} \)
73 \( 1 + 73iT^{2} \)
79 \( 1 + 17.5iT - 79T^{2} \)
83 \( 1 + (4.68 - 4.68i)T - 83iT^{2} \)
89 \( 1 - 11.0iT - 89T^{2} \)
97 \( 1 - 97iT^{2} \)
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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

−9.749826474703970949794496629039, −9.088694011795215208868149310551, −8.140892670291130958310015715427, −7.36654005863849567730307247146, −6.57599346474187409849968828086, −6.24073330212827198841771746680, −4.85761257369237698038215851078, −4.02293580801271236136853596006, −3.14984221349366030267460217296, −1.62943306527318313150579069710, 1.54206747054779494191416804502, 2.70657816864792029504999837536, 3.54578967288167025941447516813, 4.28462430855581048386975859473, 5.05970113853316881833430229198, 6.07210177322182446012014633374, 7.33534731965163309781324024787, 8.464659434257513233583919806338, 9.390899358646399410789631216551, 10.00757141388130458609164117226

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