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Properties

Label 2-2960-37.36-c1-0-74
Degree $2$
Conductor $2960$
Sign $0.164 - 0.986i$
Analytic cond. $23.6357$
Root an. cond. $4.86165$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − i·5-s − 4·7-s + 9-s − 4·11-s + 2i·13-s + 2i·15-s − 6i·17-s − 8i·19-s + 8·21-s − 4i·23-s − 25-s + 4·27-s − 6i·29-s + 2i·31-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.447i·5-s − 1.51·7-s + 0.333·9-s − 1.20·11-s + 0.554i·13-s + 0.516i·15-s − 1.45i·17-s − 1.83i·19-s + 1.74·21-s − 0.834i·23-s − 0.200·25-s + 0.769·27-s − 1.11i·29-s + 0.359i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.164 - 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.164 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2960\)    =    \(2^{4} \cdot 5 \cdot 37\)
Sign: $0.164 - 0.986i$
Analytic conductor: \(23.6357\)
Root analytic conductor: \(4.86165\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2960} (961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 2960,\ (\ :1/2),\ 0.164 - 0.986i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + iT \)
37 \( 1 + (6 + i)T \)
good3 \( 1 + 2T + 3T^{2} \)
7 \( 1 + 4T + 7T^{2} \)
11 \( 1 + 4T + 11T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
17 \( 1 + 6iT - 17T^{2} \)
19 \( 1 + 8iT - 19T^{2} \)
23 \( 1 + 4iT - 23T^{2} \)
29 \( 1 + 6iT - 29T^{2} \)
31 \( 1 - 2iT - 31T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 + 12T + 47T^{2} \)
53 \( 1 + 12T + 53T^{2} \)
59 \( 1 - 8iT - 59T^{2} \)
61 \( 1 - 2iT - 61T^{2} \)
67 \( 1 - 2T + 67T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 - 6T + 73T^{2} \)
79 \( 1 - 6iT - 79T^{2} \)
83 \( 1 + 14T + 83T^{2} \)
89 \( 1 - 4iT - 89T^{2} \)
97 \( 1 + 2iT - 97T^{2} \)
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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

−8.195963496082392210817032230939, −6.95548834897054889032025069250, −6.76897504079827932627861443994, −5.83025749595855266463253956511, −5.05434549987471197418732347939, −4.58460333809737277690090221391, −3.16508576882184256037017611422, −2.47386393871341137435290009143, −0.54086001010203722574803293760, 0, 1.69988502084614353525115366949, 3.19989780211323511628277880563, 3.52810544361093495382902637396, 4.96177448389323099331441841445, 5.72766927420182055937306074845, 6.16863225822711250451883460596, 6.79266619739544094292836596918, 7.82421705741254397683851130527, 8.391616309813893196866785704244

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