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Properties

Label 2-975-195.44-c1-0-70
Degree $2$
Conductor $975$
Sign $0.992 + 0.120i$
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  + (0.455 − 0.455i)2-s + (1.67 + 0.446i)3-s + 1.58i·4-s + (0.966 − 0.559i)6-s + (2.89 − 2.89i)7-s + (1.63 + 1.63i)8-s + (2.60 + 1.49i)9-s + (2.45 − 2.45i)11-s + (−0.707 + 2.65i)12-s + (−1.09 − 3.43i)13-s − 2.64i·14-s − 1.68·16-s − 6.48i·17-s + (1.86 − 0.504i)18-s + (−3.48 + 3.48i)19-s + ⋯
L(s)  = 1  + (0.322 − 0.322i)2-s + (0.966 + 0.257i)3-s + 0.792i·4-s + (0.394 − 0.228i)6-s + (1.09 − 1.09i)7-s + (0.577 + 0.577i)8-s + (0.867 + 0.498i)9-s + (0.741 − 0.741i)11-s + (−0.204 + 0.765i)12-s + (−0.304 − 0.952i)13-s − 0.705i·14-s − 0.420·16-s − 1.57i·17-s + (0.439 − 0.118i)18-s + (−0.798 + 0.798i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.120i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 + 0.120i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $0.992 + 0.120i$
Analytic conductor: \(7.78541\)
Root analytic conductor: \(2.79023\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (824, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :1/2),\ 0.992 + 0.120i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.03958 - 0.183871i\)
\(L(\frac12)\) \(\approx\) \(3.03958 - 0.183871i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.67 - 0.446i)T \)
5 \( 1 \)
13 \( 1 + (1.09 + 3.43i)T \)
good2 \( 1 + (-0.455 + 0.455i)T - 2iT^{2} \)
7 \( 1 + (-2.89 + 2.89i)T - 7iT^{2} \)
11 \( 1 + (-2.45 + 2.45i)T - 11iT^{2} \)
17 \( 1 + 6.48iT - 17T^{2} \)
19 \( 1 + (3.48 - 3.48i)T - 19iT^{2} \)
23 \( 1 - 2.39iT - 23T^{2} \)
29 \( 1 + 0.0728iT - 29T^{2} \)
31 \( 1 + (3.16 - 3.16i)T - 31iT^{2} \)
37 \( 1 + (6.34 - 6.34i)T - 37iT^{2} \)
41 \( 1 + (2.12 + 2.12i)T + 41iT^{2} \)
43 \( 1 + 3.53T + 43T^{2} \)
47 \( 1 + (-8.05 - 8.05i)T + 47iT^{2} \)
53 \( 1 - 9.32T + 53T^{2} \)
59 \( 1 + (3.31 - 3.31i)T - 59iT^{2} \)
61 \( 1 + 1.29T + 61T^{2} \)
67 \( 1 + (-0.922 - 0.922i)T + 67iT^{2} \)
71 \( 1 + (2.96 + 2.96i)T + 71iT^{2} \)
73 \( 1 + (5.91 - 5.91i)T - 73iT^{2} \)
79 \( 1 + 9.54T + 79T^{2} \)
83 \( 1 + (2.62 - 2.62i)T - 83iT^{2} \)
89 \( 1 + (7.03 - 7.03i)T - 89iT^{2} \)
97 \( 1 + (-6.77 - 6.77i)T + 97iT^{2} \)
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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

−10.12916967721619384469306652921, −8.934015218314999962241610095452, −8.325096516363907874177629159486, −7.57177425543230731723740868824, −7.04501166765804787475106513481, −5.26650550054350840238924708566, −4.38878074060024888217532447103, −3.65270551237005003203820326620, −2.80033533770378380965427526018, −1.44179943053198897365500145656, 1.76937354678321903987767792408, 2.09917243446003607012439786220, 4.02645000391457068877892915508, 4.62818398064472765950192196374, 5.75608205867177662800472353979, 6.67886441431279602817089328869, 7.37011487500341992106510435351, 8.729484326708764629273955846617, 8.808799722143082437414446128418, 9.888505958802593060059857527983

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