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Properties

Label 2-960-192.35-c1-0-124
Degree $2$
Conductor $960$
Sign $-0.346 - 0.938i$
Analytic cond. $7.66563$
Root an. cond. $2.76868$
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  + (0.831 − 1.14i)2-s + (−1.72 + 0.176i)3-s + (−0.615 − 1.90i)4-s + (−0.831 − 0.555i)5-s + (−1.23 + 2.11i)6-s + (−1.53 − 3.69i)7-s + (−2.68 − 0.878i)8-s + (2.93 − 0.607i)9-s + (−1.32 + 0.488i)10-s + (−0.519 − 2.61i)11-s + (1.39 + 3.17i)12-s + (2.12 + 3.18i)13-s + (−5.49 − 1.32i)14-s + (1.53 + 0.810i)15-s + (−3.24 + 2.34i)16-s + (−1.81 + 1.81i)17-s + ⋯
L(s)  = 1  + (0.588 − 0.808i)2-s + (−0.994 + 0.101i)3-s + (−0.307 − 0.951i)4-s + (−0.371 − 0.248i)5-s + (−0.502 + 0.864i)6-s + (−0.578 − 1.39i)7-s + (−0.950 − 0.310i)8-s + (0.979 − 0.202i)9-s + (−0.419 + 0.154i)10-s + (−0.156 − 0.787i)11-s + (0.403 + 0.915i)12-s + (0.590 + 0.883i)13-s + (−1.46 − 0.353i)14-s + (0.395 + 0.209i)15-s + (−0.810 + 0.585i)16-s + (−0.439 + 0.439i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.346 - 0.938i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.346 - 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $-0.346 - 0.938i$
Analytic conductor: \(7.66563\)
Root analytic conductor: \(2.76868\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{960} (611, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :1/2),\ -0.346 - 0.938i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.269077 + 0.386235i\)
\(L(\frac12)\) \(\approx\) \(0.269077 + 0.386235i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.831 + 1.14i)T \)
3 \( 1 + (1.72 - 0.176i)T \)
5 \( 1 + (0.831 + 0.555i)T \)
good7 \( 1 + (1.53 + 3.69i)T + (-4.94 + 4.94i)T^{2} \)
11 \( 1 + (0.519 + 2.61i)T + (-10.1 + 4.20i)T^{2} \)
13 \( 1 + (-2.12 - 3.18i)T + (-4.97 + 12.0i)T^{2} \)
17 \( 1 + (1.81 - 1.81i)T - 17iT^{2} \)
19 \( 1 + (2.62 + 3.93i)T + (-7.27 + 17.5i)T^{2} \)
23 \( 1 + (-0.157 + 0.379i)T + (-16.2 - 16.2i)T^{2} \)
29 \( 1 + (7.83 + 1.55i)T + (26.7 + 11.0i)T^{2} \)
31 \( 1 - 6.58T + 31T^{2} \)
37 \( 1 + (-6.24 - 4.17i)T + (14.1 + 34.1i)T^{2} \)
41 \( 1 + (0.724 + 0.300i)T + (28.9 + 28.9i)T^{2} \)
43 \( 1 + (-2.05 - 10.3i)T + (-39.7 + 16.4i)T^{2} \)
47 \( 1 + (-3.53 - 3.53i)T + 47iT^{2} \)
53 \( 1 + (13.8 - 2.75i)T + (48.9 - 20.2i)T^{2} \)
59 \( 1 + (-3.36 + 5.03i)T + (-22.5 - 54.5i)T^{2} \)
61 \( 1 + (8.78 + 1.74i)T + (56.3 + 23.3i)T^{2} \)
67 \( 1 + (0.123 - 0.621i)T + (-61.8 - 25.6i)T^{2} \)
71 \( 1 + (6.97 - 2.89i)T + (50.2 - 50.2i)T^{2} \)
73 \( 1 + (6.92 + 2.86i)T + (51.6 + 51.6i)T^{2} \)
79 \( 1 + (12.0 + 12.0i)T + 79iT^{2} \)
83 \( 1 + (-1.02 + 0.681i)T + (31.7 - 76.6i)T^{2} \)
89 \( 1 + (-12.8 + 5.33i)T + (62.9 - 62.9i)T^{2} \)
97 \( 1 + 6.06iT - 97T^{2} \)
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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

−9.758269747046429918935272731667, −8.938006470704479455380164713568, −7.58450350880959479536733775260, −6.43371015123552007804397691080, −6.11424340563074777538725313353, −4.54793341685083076090452342900, −4.30444204546816894538320198837, −3.21538363865552867619338116504, −1.36283415328448917473044932495, −0.21468814172592509512228454168, 2.37668471457557457594082975689, 3.65113000738186243778750161444, 4.70939042173536880133572663121, 5.70331550459846647367576701068, 6.07127956217707089450389014231, 7.05011719621223256653618787539, 7.83441377437862345054419569951, 8.801548697444145397687440496520, 9.704192089894325900684348227260, 10.73856508533077626708488894393

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