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Properties

Label 2-975-13.9-c1-0-26
Degree $2$
Conductor $975$
Sign $0.0128 + 0.999i$
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 − 1.73i)2-s + (0.5 − 0.866i)3-s + (−0.999 − 1.73i)4-s + (−0.999 − 1.73i)6-s + (2.5 + 4.33i)7-s + (−0.499 − 0.866i)9-s + (−1 + 1.73i)11-s − 1.99·12-s + (2.5 − 2.59i)13-s + 10·14-s + (1.99 − 3.46i)16-s + (1 + 1.73i)17-s − 1.99·18-s + 5·21-s + (1.99 + 3.46i)22-s + (3 − 5.19i)23-s + ⋯
L(s)  = 1  + (0.707 − 1.22i)2-s + (0.288 − 0.499i)3-s + (−0.499 − 0.866i)4-s + (−0.408 − 0.707i)6-s + (0.944 + 1.63i)7-s + (−0.166 − 0.288i)9-s + (−0.301 + 0.522i)11-s − 0.577·12-s + (0.693 − 0.720i)13-s + 2.67·14-s + (0.499 − 0.866i)16-s + (0.242 + 0.420i)17-s − 0.471·18-s + 1.09·21-s + (0.426 + 0.738i)22-s + (0.625 − 1.08i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0128 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0128 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $0.0128 + 0.999i$
Analytic conductor: \(7.78541\)
Root analytic conductor: \(2.79023\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :1/2),\ 0.0128 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.10417 - 2.07736i\)
\(L(\frac12)\) \(\approx\) \(2.10417 - 2.07736i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 \)
13 \( 1 + (-2.5 + 2.59i)T \)
good2 \( 1 + (-1 + 1.73i)T + (-1 - 1.73i)T^{2} \)
7 \( 1 + (-2.5 - 4.33i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1 - 1.73i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1 - 1.73i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3 + 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2 + 3.46i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 7T + 31T^{2} \)
37 \( 1 + (1 - 1.73i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3 - 5.19i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 8T + 47T^{2} \)
53 \( 1 - 4T + 53T^{2} \)
59 \( 1 + (6 + 10.3i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.5 - 11.2i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.5 - 6.06i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (6 + 10.3i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 15T + 73T^{2} \)
79 \( 1 - 3T + 79T^{2} \)
83 \( 1 + 8T + 83T^{2} \)
89 \( 1 + (7 - 12.1i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.5 + 4.33i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.06129316957753086885021350471, −8.909474866293837622326527584947, −8.317201609926387403092252443598, −7.43343234387023088586316125450, −6.00924555963716943950519564800, −5.28621929717501118482480550877, −4.37061448652907077986563926366, −3.06367625118072500175515010588, −2.35636734987625701627620210026, −1.43869885505388164874787171921, 1.43019433965512098771345422451, 3.52007807534941939497845543205, 4.15663118191157555751207607762, 5.01430051905080101940031568995, 5.76168188006061868435692921764, 7.10806293785033928768809006999, 7.34346679751807925245553439464, 8.314016588961537754189340428512, 9.107452310553927352648300173741, 10.43049998549944312341152511767

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