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Properties

Label 2-975-65.9-c1-0-32
Degree $2$
Conductor $975$
Sign $0.450 - 0.892i$
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  + (2.21 + 1.28i)2-s + (−0.866 − 0.5i)3-s + (2.28 + 3.95i)4-s + (−1.28 − 2.21i)6-s + (3.08 − 1.78i)7-s + 6.56i·8-s + (0.499 + 0.866i)9-s + (1 − 1.73i)11-s − 4.56i·12-s + (3.57 + 0.5i)13-s + 9.12·14-s + (−3.84 + 6.65i)16-s + (−2.21 + 1.28i)17-s + 2.56i·18-s + (−0.561 − 0.972i)19-s + ⋯
L(s)  = 1  + (1.56 + 0.905i)2-s + (−0.499 − 0.288i)3-s + (1.14 + 1.97i)4-s + (−0.522 − 0.905i)6-s + (1.16 − 0.673i)7-s + 2.31i·8-s + (0.166 + 0.288i)9-s + (0.301 − 0.522i)11-s − 1.31i·12-s + (0.990 + 0.138i)13-s + 2.43·14-s + (−0.960 + 1.66i)16-s + (−0.538 + 0.310i)17-s + 0.603i·18-s + (−0.128 − 0.223i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.450 - 0.892i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.450 - 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $0.450 - 0.892i$
Analytic conductor: \(7.78541\)
Root analytic conductor: \(2.79023\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (724, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :1/2),\ 0.450 - 0.892i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.23921 + 1.99380i\)
\(L(\frac12)\) \(\approx\) \(3.23921 + 1.99380i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.866 + 0.5i)T \)
5 \( 1 \)
13 \( 1 + (-3.57 - 0.5i)T \)
good2 \( 1 + (-2.21 - 1.28i)T + (1 + 1.73i)T^{2} \)
7 \( 1 + (-3.08 + 1.78i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1 + 1.73i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (2.21 - 1.28i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.561 + 0.972i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.73 + i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.84 - 4.92i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 1.56T + 31T^{2} \)
37 \( 1 + (-2.97 - 1.71i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.28 - 2.21i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.379 + 0.219i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 8.24iT - 47T^{2} \)
53 \( 1 - 11.6iT - 53T^{2} \)
59 \( 1 + (5.56 + 9.63i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.06 + 10.4i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.379 - 0.219i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (7 + 12.1i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 1.87iT - 73T^{2} \)
79 \( 1 + 9.56T + 79T^{2} \)
83 \( 1 + 9.12iT - 83T^{2} \)
89 \( 1 + (-6.56 + 11.3i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.84 + 2.21i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.74960932783229664605515057565, −8.960256857157523890267652242431, −8.010350929940172427547605295191, −7.43131096360612999864611153862, −6.41662221913434678374831985197, −5.95152448154802315811676070114, −4.84052266981001478327102447705, −4.30426859820601916826247569271, −3.27673836410607749528869622382, −1.60360467727692540339279648819, 1.48692520106216810671316459752, 2.42242174595223865497055331108, 3.82533939277645362843121951178, 4.43171502086696322807791983433, 5.38143042501547737118817396027, 5.87587823253042500150131451295, 6.91697214075337980506260824994, 8.243162938039922743841372916094, 9.305147586573910571060499595304, 10.33406379978787693962128654566

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