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Properties

Label 2-975-195.164-c1-0-70
Degree $2$
Conductor $975$
Sign $0.602 + 0.798i$
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  + (1.57 + 1.57i)2-s + (−0.0822 − 1.73i)3-s + 2.98i·4-s + (2.60 − 2.86i)6-s + (−2.29 − 2.29i)7-s + (−1.56 + 1.56i)8-s + (−2.98 + 0.284i)9-s + (−3.85 − 3.85i)11-s + (5.16 − 0.245i)12-s + (3.52 + 0.766i)13-s − 7.24i·14-s + 1.04·16-s − 3.78i·17-s + (−5.16 − 4.26i)18-s + (1.28 + 1.28i)19-s + ⋯
L(s)  = 1  + (1.11 + 1.11i)2-s + (−0.0474 − 0.998i)3-s + 1.49i·4-s + (1.06 − 1.16i)6-s + (−0.866 − 0.866i)7-s + (−0.551 + 0.551i)8-s + (−0.995 + 0.0948i)9-s + (−1.16 − 1.16i)11-s + (1.49 − 0.0709i)12-s + (0.977 + 0.212i)13-s − 1.93i·14-s + 0.261·16-s − 0.917i·17-s + (−1.21 − 1.00i)18-s + (0.295 + 0.295i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.602 + 0.798i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.602 + 0.798i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $0.602 + 0.798i$
Analytic conductor: \(7.78541\)
Root analytic conductor: \(2.79023\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (749, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :1/2),\ 0.602 + 0.798i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.86183 - 0.927431i\)
\(L(\frac12)\) \(\approx\) \(1.86183 - 0.927431i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.0822 + 1.73i)T \)
5 \( 1 \)
13 \( 1 + (-3.52 - 0.766i)T \)
good2 \( 1 + (-1.57 - 1.57i)T + 2iT^{2} \)
7 \( 1 + (2.29 + 2.29i)T + 7iT^{2} \)
11 \( 1 + (3.85 + 3.85i)T + 11iT^{2} \)
17 \( 1 + 3.78iT - 17T^{2} \)
19 \( 1 + (-1.28 - 1.28i)T + 19iT^{2} \)
23 \( 1 + 5.74iT - 23T^{2} \)
29 \( 1 + 5.42iT - 29T^{2} \)
31 \( 1 + (3.95 + 3.95i)T + 31iT^{2} \)
37 \( 1 + (-6.99 - 6.99i)T + 37iT^{2} \)
41 \( 1 + (0.872 - 0.872i)T - 41iT^{2} \)
43 \( 1 + 4.18T + 43T^{2} \)
47 \( 1 + (1.56 - 1.56i)T - 47iT^{2} \)
53 \( 1 - 1.55T + 53T^{2} \)
59 \( 1 + (4.00 + 4.00i)T + 59iT^{2} \)
61 \( 1 + 2.47T + 61T^{2} \)
67 \( 1 + (5.37 - 5.37i)T - 67iT^{2} \)
71 \( 1 + (1.46 - 1.46i)T - 71iT^{2} \)
73 \( 1 + (-5.34 - 5.34i)T + 73iT^{2} \)
79 \( 1 - 4.01T + 79T^{2} \)
83 \( 1 + (-3.77 - 3.77i)T + 83iT^{2} \)
89 \( 1 + (-4.28 - 4.28i)T + 89iT^{2} \)
97 \( 1 + (-12.3 + 12.3i)T - 97iT^{2} \)
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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

−9.887812814539988805888591969979, −8.511698812968411720889132598926, −7.88302098750010576902918683086, −7.15516358613787875172006732827, −6.27596308463558719238766686344, −5.94360996086580841656261346590, −4.83359272870548047901132746712, −3.62186413893676332620620371774, −2.81341871494182400352857509602, −0.65380567971169262994871648301, 1.98500479605423649626151689090, 3.08184737068033424268275829176, 3.63920236076491946438994375541, 4.76393183031971283858659334137, 5.46524972483795497809489703417, 6.12688198050800558466887747561, 7.67200401185880938794226154522, 8.880779891911120843590100793208, 9.596019643702995935303913474307, 10.43081476089604997522779015003

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