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Properties

Label 2-975-195.89-c1-0-60
Degree $2$
Conductor $975$
Sign $-0.735 + 0.677i$
Analytic cond. $7.78541$
Root an. cond. $2.79023$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.5 + 0.866i)3-s + (1.73 + i)4-s + (−2.86 + 0.767i)7-s + (1.5 − 2.59i)9-s − 3.46·12-s + (−2.5 − 2.59i)13-s + (1.99 + 3.46i)16-s + (−7.83 + 2.09i)19-s + (3.63 − 3.63i)21-s + 5.19i·27-s + (−5.73 − 1.53i)28-s + (−7.83 − 7.83i)31-s + (5.19 − 3i)36-s + (0.562 − 2.09i)37-s + (6 + 1.73i)39-s + ⋯
L(s)  = 1  + (−0.866 + 0.499i)3-s + (0.866 + 0.5i)4-s + (−1.08 + 0.290i)7-s + (0.5 − 0.866i)9-s − 0.999·12-s + (−0.693 − 0.720i)13-s + (0.499 + 0.866i)16-s + (−1.79 + 0.481i)19-s + (0.792 − 0.792i)21-s + 0.999i·27-s + (−1.08 − 0.290i)28-s + (−1.40 − 1.40i)31-s + (0.866 − 0.5i)36-s + (0.0924 − 0.344i)37-s + (0.960 + 0.277i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.735 + 0.677i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.735 + 0.677i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $-0.735 + 0.677i$
Analytic conductor: \(7.78541\)
Root analytic conductor: \(2.79023\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (674, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 975,\ (\ :1/2),\ -0.735 + 0.677i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.5 - 0.866i)T \)
5 \( 1 \)
13 \( 1 + (2.5 + 2.59i)T \)
good2 \( 1 + (-1.73 - i)T^{2} \)
7 \( 1 + (2.86 - 0.767i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (7.83 - 2.09i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (7.83 + 7.83i)T + 31iT^{2} \)
37 \( 1 + (-0.562 + 2.09i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-0.866 + 1.5i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 47iT^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-4.33 + 7.5i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.767 + 0.205i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (9.36 + 9.36i)T + 73iT^{2} \)
79 \( 1 + 12.1T + 79T^{2} \)
83 \( 1 + 83iT^{2} \)
89 \( 1 + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-4.40 - 16.4i)T + (-84.0 + 48.5i)T^{2} \)
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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

−9.890051698805383605611469382914, −9.012834820996547957191624480812, −7.87654328670063449819388349233, −6.96042672102577887323300339250, −6.20647809448005555199602351692, −5.61081777659943340123822374157, −4.23446530904261523206521280885, −3.35875859745142699770047965140, −2.19518751253404463933572604832, 0, 1.65087472993228897168240014104, 2.73399390597629096378937745519, 4.24942888481424370828395459724, 5.35006408389222448937739538485, 6.27163298059165830853642416618, 6.84495696650333392851829935713, 7.31637318127687022471457338785, 8.695647111891489144920275236388, 9.792447155691993141290983910409

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