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Properties

Label 2-975-975.779-c0-0-1
Degree $2$
Conductor $975$
Sign $-0.248 - 0.968i$
Analytic cond. $0.486588$
Root an. cond. $0.697558$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.69 + 0.550i)2-s + (−0.587 + 0.809i)3-s + (1.76 + 1.27i)4-s + (−0.156 + 0.987i)5-s + (−1.44 + 1.04i)6-s + (1.23 + 1.69i)8-s + (−0.309 − 0.951i)9-s + (−0.809 + 1.58i)10-s + (0.610 − 1.87i)11-s + (−2.06 + 0.672i)12-s + (−0.951 + 0.309i)13-s + (−0.707 − 0.707i)15-s + (0.481 + 1.48i)16-s − 1.78i·18-s + (−1.53 + 1.53i)20-s + ⋯
L(s)  = 1  + (1.69 + 0.550i)2-s + (−0.587 + 0.809i)3-s + (1.76 + 1.27i)4-s + (−0.156 + 0.987i)5-s + (−1.44 + 1.04i)6-s + (1.23 + 1.69i)8-s + (−0.309 − 0.951i)9-s + (−0.809 + 1.58i)10-s + (0.610 − 1.87i)11-s + (−2.06 + 0.672i)12-s + (−0.951 + 0.309i)13-s + (−0.707 − 0.707i)15-s + (0.481 + 1.48i)16-s − 1.78i·18-s + (−1.53 + 1.53i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.248 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.248 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $-0.248 - 0.968i$
Analytic conductor: \(0.486588\)
Root analytic conductor: \(0.697558\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (779, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :0),\ -0.248 - 0.968i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.067406410\)
\(L(\frac12)\) \(\approx\) \(2.067406410\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.587 - 0.809i)T \)
5 \( 1 + (0.156 - 0.987i)T \)
13 \( 1 + (0.951 - 0.309i)T \)
good2 \( 1 + (-1.69 - 0.550i)T + (0.809 + 0.587i)T^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 + (-0.610 + 1.87i)T + (-0.809 - 0.587i)T^{2} \)
17 \( 1 + (0.309 - 0.951i)T^{2} \)
19 \( 1 + (-0.309 + 0.951i)T^{2} \)
23 \( 1 + (-0.809 - 0.587i)T^{2} \)
29 \( 1 + (-0.309 - 0.951i)T^{2} \)
31 \( 1 + (-0.309 + 0.951i)T^{2} \)
37 \( 1 + (-0.809 + 0.587i)T^{2} \)
41 \( 1 + (0.280 + 0.863i)T + (-0.809 + 0.587i)T^{2} \)
43 \( 1 - 1.90iT - T^{2} \)
47 \( 1 + (-0.533 + 0.734i)T + (-0.309 - 0.951i)T^{2} \)
53 \( 1 + (0.309 + 0.951i)T^{2} \)
59 \( 1 + (-0.0966 - 0.297i)T + (-0.809 + 0.587i)T^{2} \)
61 \( 1 + (0.363 - 1.11i)T + (-0.809 - 0.587i)T^{2} \)
67 \( 1 + (0.309 - 0.951i)T^{2} \)
71 \( 1 + (-1.59 - 1.16i)T + (0.309 + 0.951i)T^{2} \)
73 \( 1 + (-0.809 - 0.587i)T^{2} \)
79 \( 1 + (0.309 + 0.951i)T^{2} \)
83 \( 1 + (1.04 + 1.44i)T + (-0.309 + 0.951i)T^{2} \)
89 \( 1 + (-0.437 + 1.34i)T + (-0.809 - 0.587i)T^{2} \)
97 \( 1 + (0.309 + 0.951i)T^{2} \)
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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

−10.84062202365955853932302979646, −9.860361475051690263649881224634, −8.684909799359164703912761304461, −7.49531823135230553339974254862, −6.57671421246473260614844548346, −6.06925298686596196173907349042, −5.29577974532296372989422515952, −4.26642474357832526481043596457, −3.50646175520195044645420880992, −2.79402913931028338984607056917, 1.56530442872749372626114935026, 2.38303108319699683369234084055, 3.95545826201977215260031525426, 4.89213715993369115895029339401, 5.17826571084999592092588874553, 6.36548873883969120230120221708, 7.09930055672330924424868958408, 7.940047601728098962539063414985, 9.408431583534543739753173966471, 10.23653774451967240455158553167

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