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Properties

Label 2-975-39.11-c1-0-5
Degree $2$
Conductor $975$
Sign $-0.936 + 0.351i$
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  + (1.45 + 0.389i)2-s + (−0.239 + 1.71i)3-s + (0.232 + 0.133i)4-s + (−1.01 + 2.40i)6-s + (−0.366 − 1.36i)7-s + (−1.84 − 1.84i)8-s + (−2.88 − 0.820i)9-s + (−1.06 + 3.97i)11-s + (−0.285 + 0.366i)12-s + (−3.59 − 0.232i)13-s − 2.12i·14-s + (−2.23 − 3.86i)16-s + (−2.51 + 4.36i)17-s + (−3.87 − 2.31i)18-s + (−3.73 + i)19-s + ⋯
L(s)  = 1  + (1.02 + 0.275i)2-s + (−0.138 + 0.990i)3-s + (0.116 + 0.0669i)4-s + (−0.415 + 0.980i)6-s + (−0.138 − 0.516i)7-s + (−0.652 − 0.652i)8-s + (−0.961 − 0.273i)9-s + (−0.321 + 1.19i)11-s + (−0.0823 + 0.105i)12-s + (−0.997 − 0.0643i)13-s − 0.569i·14-s + (−0.558 − 0.966i)16-s + (−0.611 + 1.05i)17-s + (−0.913 − 0.546i)18-s + (−0.856 + 0.229i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.936 + 0.351i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.936 + 0.351i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $-0.936 + 0.351i$
Analytic conductor: \(7.78541\)
Root analytic conductor: \(2.79023\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (401, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :1/2),\ -0.936 + 0.351i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.106747 - 0.587852i\)
\(L(\frac12)\) \(\approx\) \(0.106747 - 0.587852i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.239 - 1.71i)T \)
5 \( 1 \)
13 \( 1 + (3.59 + 0.232i)T \)
good2 \( 1 + (-1.45 - 0.389i)T + (1.73 + i)T^{2} \)
7 \( 1 + (0.366 + 1.36i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (1.06 - 3.97i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (2.51 - 4.36i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.73 - i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (6.20 - 3.58i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.46 + 2.46i)T + 31iT^{2} \)
37 \( 1 + (-5.23 - 1.40i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-5.42 - 1.45i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (1.90 + 1.09i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.25 - 4.25i)T + 47iT^{2} \)
53 \( 1 - 0.779iT - 53T^{2} \)
59 \( 1 + (2.90 - 0.779i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-3.5 + 6.06i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.53 + 5.73i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (0.779 + 2.90i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (-0.901 + 0.901i)T - 73iT^{2} \)
79 \( 1 - 2T + 79T^{2} \)
83 \( 1 + (2.90 - 2.90i)T - 83iT^{2} \)
89 \( 1 + (2.41 - 9.01i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (1.63 - 0.437i)T + (84.0 - 48.5i)T^{2} \)
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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

−10.42275819472472312406703328753, −9.697530894566653547112349366800, −9.091850891240212129582812794833, −7.82988668378815303433970063662, −6.83062269324277059046790798604, −5.93994896364413066267908055343, −5.04660639727060660063822821513, −4.33124950539806294901826258582, −3.76996122639290266620051768757, −2.39304220977174885197709535064, 0.18250054837887662791550716653, 2.33025544697962046047301619787, 2.85586201712206998965326900394, 4.19372127505751613577308808879, 5.34459098775416035224956372268, 5.81903984804242908054890364617, 6.82751637185268056062617839057, 7.78758872800951143629918538859, 8.704317936391059218070467424975, 9.301901135681487095210994178195

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