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Properties

Label 2-960-15.14-c0-0-0
Degree $2$
Conductor $960$
Sign $-0.707 - 0.707i$
Analytic cond. $0.479102$
Root an. cond. $0.692172$
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  + (−0.707 + 0.707i)3-s + i·5-s + 1.41i·7-s − 1.00i·9-s + (−0.707 − 0.707i)15-s + (−1.00 − 1.00i)21-s − 1.41·23-s − 25-s + (0.707 + 0.707i)27-s − 1.41·35-s + 2i·41-s − 1.41i·43-s + 1.00·45-s + 1.41·47-s − 1.00·49-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)3-s + i·5-s + 1.41i·7-s − 1.00i·9-s + (−0.707 − 0.707i)15-s + (−1.00 − 1.00i)21-s − 1.41·23-s − 25-s + (0.707 + 0.707i)27-s − 1.41·35-s + 2i·41-s − 1.41i·43-s + 1.00·45-s + 1.41·47-s − 1.00·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $-0.707 - 0.707i$
Analytic conductor: \(0.479102\)
Root analytic conductor: \(0.692172\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{960} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :0),\ -0.707 - 0.707i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6955568829\)
\(L(\frac12)\) \(\approx\) \(0.6955568829\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 - iT \)
good7 \( 1 - 1.41iT - T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + 1.41T + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - 2iT - T^{2} \)
43 \( 1 + 1.41iT - T^{2} \)
47 \( 1 - 1.41T + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 - 1.41iT - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - 1.41T + T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - T^{2} \)
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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

−10.48126833242468539348425655755, −9.852220340232146720721451859201, −9.060588234949771177859009496019, −8.126281039393325144073570180057, −6.94813443501215292307110206074, −6.02686555939227417098398145403, −5.60187859969597359570905049875, −4.37929042112545170033424827799, −3.30311515409524620507614246061, −2.23925639839707580337545447750, 0.73426136073904767600260083444, 1.95048848839723378219478265050, 3.85642176500160888762912769590, 4.64665948242634331572115219844, 5.62535537150994476760491046564, 6.49810052096633444770319564646, 7.50964584855919980470988815942, 7.953636877011997251926841059475, 9.052182571376910043620438389432, 10.12040533344124617888249023222

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