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Properties

Label 2-975-39.5-c1-0-5
Degree $2$
Conductor $975$
Sign $0.712 - 0.702i$
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  + (−1.89 − 1.89i)2-s + (−0.329 − 1.70i)3-s + 5.15i·4-s + (−2.59 + 3.83i)6-s + (3.13 + 3.13i)7-s + (5.96 − 5.96i)8-s + (−2.78 + 1.12i)9-s + (−1.22 + 1.22i)11-s + (8.76 − 1.70i)12-s + (−1.04 − 3.45i)13-s − 11.8i·14-s − 12.2·16-s − 2.34·17-s + (7.38 + 3.13i)18-s + (1.45 − 1.45i)19-s + ⋯
L(s)  = 1  + (−1.33 − 1.33i)2-s + (−0.190 − 0.981i)3-s + 2.57i·4-s + (−1.05 + 1.56i)6-s + (1.18 + 1.18i)7-s + (2.10 − 2.10i)8-s + (−0.927 + 0.374i)9-s + (−0.368 + 0.368i)11-s + (2.52 − 0.490i)12-s + (−0.288 − 0.957i)13-s − 3.17i·14-s − 3.06·16-s − 0.567·17-s + (1.74 + 0.739i)18-s + (0.333 − 0.333i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.712 - 0.702i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.712 - 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $0.712 - 0.702i$
Analytic conductor: \(7.78541\)
Root analytic conductor: \(2.79023\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (551, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :1/2),\ 0.712 - 0.702i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.201029 + 0.0824455i\)
\(L(\frac12)\) \(\approx\) \(0.201029 + 0.0824455i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.329 + 1.70i)T \)
5 \( 1 \)
13 \( 1 + (1.04 + 3.45i)T \)
good2 \( 1 + (1.89 + 1.89i)T + 2iT^{2} \)
7 \( 1 + (-3.13 - 3.13i)T + 7iT^{2} \)
11 \( 1 + (1.22 - 1.22i)T - 11iT^{2} \)
17 \( 1 + 2.34T + 17T^{2} \)
19 \( 1 + (-1.45 + 1.45i)T - 19iT^{2} \)
23 \( 1 + 7.03T + 23T^{2} \)
29 \( 1 - 4.84iT - 29T^{2} \)
31 \( 1 + (4.83 - 4.83i)T - 31iT^{2} \)
37 \( 1 + (1.45 + 1.45i)T + 37iT^{2} \)
41 \( 1 + (-0.517 - 0.517i)T + 41iT^{2} \)
43 \( 1 + 5.47iT - 43T^{2} \)
47 \( 1 + (1.64 - 1.64i)T - 47iT^{2} \)
53 \( 1 - 5.90iT - 53T^{2} \)
59 \( 1 + (8.44 - 8.44i)T - 59iT^{2} \)
61 \( 1 + 6.61T + 61T^{2} \)
67 \( 1 + (8.18 - 8.18i)T - 67iT^{2} \)
71 \( 1 + (3.16 + 3.16i)T + 71iT^{2} \)
73 \( 1 + (9.62 + 9.62i)T + 73iT^{2} \)
79 \( 1 - 2.87T + 79T^{2} \)
83 \( 1 + (0.830 + 0.830i)T + 83iT^{2} \)
89 \( 1 + (-11.9 + 11.9i)T - 89iT^{2} \)
97 \( 1 + (5.59 - 5.59i)T - 97iT^{2} \)
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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

−10.36845638556460958149907521417, −9.019297221704520310874632672888, −8.694630781487557804674422149145, −7.75815318276499387091103057086, −7.39462178897770164768749792707, −5.84879792061822732091400001465, −4.78637964722991037262542432660, −3.07832750263660690053315223772, −2.22439558120732711087862055471, −1.50334363008209987816418990343, 0.16000968633741620801274629867, 1.79356057045455530817742651701, 4.10462740364750435767468297264, 4.80874996591704215702724452376, 5.77922016769115973266207517481, 6.61736392666124718913013853994, 7.72208282941542529832192268436, 8.067974766147303526794809248324, 9.050134408576705979502288509411, 9.774614489427361323089980006961

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