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Properties

Label 2-975-195.164-c1-0-30
Degree $2$
Conductor $975$
Sign $0.145 - 0.989i$
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  + (1.43 + 1.43i)2-s + (−1.72 + 0.158i)3-s + 2.13i·4-s + (−2.70 − 2.25i)6-s + (−1.21 − 1.21i)7-s + (−0.201 + 0.201i)8-s + (2.94 − 0.546i)9-s + (0.581 + 0.581i)11-s + (−0.339 − 3.69i)12-s + (3.12 − 1.79i)13-s − 3.49i·14-s + 3.70·16-s + 2.27i·17-s + (5.03 + 3.45i)18-s + (4.21 + 4.21i)19-s + ⋯
L(s)  = 1  + (1.01 + 1.01i)2-s + (−0.995 + 0.0915i)3-s + 1.06i·4-s + (−1.10 − 0.919i)6-s + (−0.458 − 0.458i)7-s + (−0.0710 + 0.0710i)8-s + (0.983 − 0.182i)9-s + (0.175 + 0.175i)11-s + (−0.0979 − 1.06i)12-s + (0.868 − 0.496i)13-s − 0.933i·14-s + 0.925·16-s + 0.552i·17-s + (1.18 + 0.814i)18-s + (0.966 + 0.966i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.145 - 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.145 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $0.145 - 0.989i$
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: \((2,\ 975,\ (\ :1/2),\ 0.145 - 0.989i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.58814 + 1.37113i\)
\(L(\frac12)\) \(\approx\) \(1.58814 + 1.37113i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.72 - 0.158i)T \)
5 \( 1 \)
13 \( 1 + (-3.12 + 1.79i)T \)
good2 \( 1 + (-1.43 - 1.43i)T + 2iT^{2} \)
7 \( 1 + (1.21 + 1.21i)T + 7iT^{2} \)
11 \( 1 + (-0.581 - 0.581i)T + 11iT^{2} \)
17 \( 1 - 2.27iT - 17T^{2} \)
19 \( 1 + (-4.21 - 4.21i)T + 19iT^{2} \)
23 \( 1 - 3.13iT - 23T^{2} \)
29 \( 1 - 3.12iT - 29T^{2} \)
31 \( 1 + (-2.36 - 2.36i)T + 31iT^{2} \)
37 \( 1 + (7.73 + 7.73i)T + 37iT^{2} \)
41 \( 1 + (-5.66 + 5.66i)T - 41iT^{2} \)
43 \( 1 - 5.44T + 43T^{2} \)
47 \( 1 + (3.80 - 3.80i)T - 47iT^{2} \)
53 \( 1 + 2.40T + 53T^{2} \)
59 \( 1 + (-10.5 - 10.5i)T + 59iT^{2} \)
61 \( 1 - 5.20T + 61T^{2} \)
67 \( 1 + (-7.77 + 7.77i)T - 67iT^{2} \)
71 \( 1 + (5.74 - 5.74i)T - 71iT^{2} \)
73 \( 1 + (-4.15 - 4.15i)T + 73iT^{2} \)
79 \( 1 - 9.23T + 79T^{2} \)
83 \( 1 + (-3.87 - 3.87i)T + 83iT^{2} \)
89 \( 1 + (4.74 + 4.74i)T + 89iT^{2} \)
97 \( 1 + (9.51 - 9.51i)T - 97iT^{2} \)
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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

−10.35647908366811256600063295611, −9.490388161066041194056369325373, −8.141177043934153576407450357134, −7.22988293773490994835761807775, −6.66693530851218317454216734626, −5.67337175120032274606550046732, −5.39673272503025870286209829474, −4.04813655851882410282514207853, −3.58888003491420574190703201427, −1.21149306430703242230237257166, 1.02695328353871836402224520310, 2.42246854225909831370733968116, 3.50744716005544726830550742831, 4.51953744306312643539748068800, 5.24892036039711755088501666349, 6.14724131611817109251458000440, 6.88718282550910280718599362620, 8.153123913200102890629566039103, 9.377283365786491295985085839959, 10.07976793858126053272849048687

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