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Properties

Label 2-975-195.44-c1-0-35
Degree $2$
Conductor $975$
Sign $-0.323 - 0.946i$
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 + (−2.25 + 2.70i)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 + (−3.45 + 5.03i)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 + (−0.919 + 1.10i)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 + (−0.814 + 1.18i)18-s + (0.966 − 0.966i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.323 - 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.323 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $-0.323 - 0.946i$
Analytic conductor: \(7.78541\)
Root analytic conductor: \(2.79023\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (824, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :1/2),\ -0.323 - 0.946i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.735492 + 1.02907i\)
\(L(\frac12)\) \(\approx\) \(0.735492 + 1.02907i\)
\(L(\frac{3}{2})\) 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

−9.793785154581029103215398503201, −9.102880486979404574738179128408, −8.688207249478123403396660383542, −7.83557538602832793433873549258, −7.06629203154007004920123771063, −6.41712985698189617695334145263, −5.33711015896820910883690402537, −3.86080088069523171287373286084, −2.87010220446036252510619464912, −1.33118857136311357560295190597, 0.835600576113357111475792038222, 2.08180062260344553437286307622, 3.20660504772891866383148968925, 3.74705023701450576027361368463, 5.30514641981154073007324321489, 6.62112084088560024279592892243, 7.74866151106364960515927996001, 8.272772670421077852865378475521, 9.106908200010650669427781716837, 9.774512185709311309716490698897

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