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Properties

Label 2-975-195.164-c1-0-55
Degree $2$
Conductor $975$
Sign $-0.0977 + 0.995i$
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  + (−0.260 − 0.260i)2-s + (1.26 − 1.18i)3-s − 1.86i·4-s + (−0.637 − 0.0199i)6-s + (2.54 + 2.54i)7-s + (−1.00 + 1.00i)8-s + (0.187 − 2.99i)9-s + (−0.348 − 0.348i)11-s + (−2.21 − 2.35i)12-s + (3.58 + 0.339i)13-s − 1.32i·14-s − 3.20·16-s − 5.28i·17-s + (−0.828 + 0.730i)18-s + (3.44 + 3.44i)19-s + ⋯
L(s)  = 1  + (−0.184 − 0.184i)2-s + (0.728 − 0.684i)3-s − 0.932i·4-s + (−0.260 − 0.00814i)6-s + (0.961 + 0.961i)7-s + (−0.355 + 0.355i)8-s + (0.0625 − 0.998i)9-s + (−0.105 − 0.105i)11-s + (−0.638 − 0.679i)12-s + (0.995 + 0.0942i)13-s − 0.354i·14-s − 0.801·16-s − 1.28i·17-s + (−0.195 + 0.172i)18-s + (0.791 + 0.791i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0977 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0977 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $-0.0977 + 0.995i$
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: \((2,\ 975,\ (\ :1/2),\ -0.0977 + 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.38210 - 1.52455i\)
\(L(\frac12)\) \(\approx\) \(1.38210 - 1.52455i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.26 + 1.18i)T \)
5 \( 1 \)
13 \( 1 + (-3.58 - 0.339i)T \)
good2 \( 1 + (0.260 + 0.260i)T + 2iT^{2} \)
7 \( 1 + (-2.54 - 2.54i)T + 7iT^{2} \)
11 \( 1 + (0.348 + 0.348i)T + 11iT^{2} \)
17 \( 1 + 5.28iT - 17T^{2} \)
19 \( 1 + (-3.44 - 3.44i)T + 19iT^{2} \)
23 \( 1 + 9.38iT - 23T^{2} \)
29 \( 1 - 6.80iT - 29T^{2} \)
31 \( 1 + (2.96 + 2.96i)T + 31iT^{2} \)
37 \( 1 + (-1.76 - 1.76i)T + 37iT^{2} \)
41 \( 1 + (-2.21 + 2.21i)T - 41iT^{2} \)
43 \( 1 - 5.41T + 43T^{2} \)
47 \( 1 + (2.46 - 2.46i)T - 47iT^{2} \)
53 \( 1 + 6.94T + 53T^{2} \)
59 \( 1 + (0.248 + 0.248i)T + 59iT^{2} \)
61 \( 1 + 3.60T + 61T^{2} \)
67 \( 1 + (-1.34 + 1.34i)T - 67iT^{2} \)
71 \( 1 + (11.6 - 11.6i)T - 71iT^{2} \)
73 \( 1 + (6.48 + 6.48i)T + 73iT^{2} \)
79 \( 1 + 2.66T + 79T^{2} \)
83 \( 1 + (-7.35 - 7.35i)T + 83iT^{2} \)
89 \( 1 + (2.54 + 2.54i)T + 89iT^{2} \)
97 \( 1 + (10.4 - 10.4i)T - 97iT^{2} \)
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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

−9.556704825567844858102619722377, −8.888959584046246637020361413210, −8.354723401656892986576379106915, −7.36125295242781562701384087585, −6.30754293228629949925704729752, −5.57874235684842527126456074904, −4.57418502118920632170700209094, −3.00687553425184327559926896315, −2.06305866111726216641461035093, −1.04676678738155770808832057914, 1.65026124149872703372560770029, 3.19054191861940370091595866505, 3.90410651040196338191520971689, 4.64314330860708890134295832907, 5.91445999924999531173228219395, 7.40805929240718445686515701824, 7.69983284683446816705019752711, 8.505448051000956641479598405704, 9.231590907365348696727602969667, 10.13052608747723477390874556813

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