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Properties

Label 2-975-195.164-c1-0-53
Degree $2$
Conductor $975$
Sign $0.141 + 0.989i$
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  + (0.707 + 0.707i)2-s + (−1.41 − i)3-s − 0.999i·4-s + (−0.292 − 1.70i)6-s + (−1 − i)7-s + (2.12 − 2.12i)8-s + (1.00 + 2.82i)9-s + (2.82 + 2.82i)11-s + (−0.999 + 1.41i)12-s + (3 − 2i)13-s − 1.41i·14-s + 1.00·16-s + (−1.29 + 2.70i)18-s + (−1 − i)19-s + (0.414 + 2.41i)21-s + 4.00i·22-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (−0.816 − 0.577i)3-s − 0.499i·4-s + (−0.119 − 0.696i)6-s + (−0.377 − 0.377i)7-s + (0.750 − 0.750i)8-s + (0.333 + 0.942i)9-s + (0.852 + 0.852i)11-s + (−0.288 + 0.408i)12-s + (0.832 − 0.554i)13-s − 0.377i·14-s + 0.250·16-s + (−0.304 + 0.638i)18-s + (−0.229 − 0.229i)19-s + (0.0903 + 0.526i)21-s + 0.852i·22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.141 + 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.141 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $0.141 + 0.989i$
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: \((2,\ 975,\ (\ :1/2),\ 0.141 + 0.989i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.12885 - 0.978783i\)
\(L(\frac12)\) \(\approx\) \(1.12885 - 0.978783i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.41 + i)T \)
5 \( 1 \)
13 \( 1 + (-3 + 2i)T \)
good2 \( 1 + (-0.707 - 0.707i)T + 2iT^{2} \)
7 \( 1 + (1 + i)T + 7iT^{2} \)
11 \( 1 + (-2.82 - 2.82i)T + 11iT^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 + (1 + i)T + 19iT^{2} \)
23 \( 1 + 8.48iT - 23T^{2} \)
29 \( 1 - 2.82iT - 29T^{2} \)
31 \( 1 + (5 + 5i)T + 31iT^{2} \)
37 \( 1 + (1 + i)T + 37iT^{2} \)
41 \( 1 + (-1.41 + 1.41i)T - 41iT^{2} \)
43 \( 1 + 6T + 43T^{2} \)
47 \( 1 + (-2.82 + 2.82i)T - 47iT^{2} \)
53 \( 1 + 5.65T + 53T^{2} \)
59 \( 1 + (2.82 + 2.82i)T + 59iT^{2} \)
61 \( 1 - 8T + 61T^{2} \)
67 \( 1 + (5 - 5i)T - 67iT^{2} \)
71 \( 1 + (2.82 - 2.82i)T - 71iT^{2} \)
73 \( 1 + (-1 - i)T + 73iT^{2} \)
79 \( 1 - 10T + 79T^{2} \)
83 \( 1 + (-5.65 - 5.65i)T + 83iT^{2} \)
89 \( 1 + (-9.89 - 9.89i)T + 89iT^{2} \)
97 \( 1 + (-7 + 7i)T - 97iT^{2} \)
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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

−10.08588022312972337751372459419, −8.996343870742000267881830225308, −7.81433074348019573424455279264, −6.80191392963236127299069907019, −6.54543895398255541263105630406, −5.61206961679028110060161780501, −4.71797818516801333994635135675, −3.85497653691125643962498321455, −1.96101929998877069000015819085, −0.69967314444000122778498041769, 1.54947218763846432001036652910, 3.34174272300163568225809605055, 3.73931228074371656073549150969, 4.81574788053086594733739152895, 5.82633289122574682253679288012, 6.50014283613447435602752137417, 7.64667275332521907793739478296, 8.840536990656040796549792308529, 9.307788808191469348680096112692, 10.47717821410700206991541553097

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