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Properties

Label 2-1512-63.25-c1-0-5
Degree $2$
Conductor $1512$
Sign $-0.926 - 0.375i$
Analytic cond. $12.0733$
Root an. cond. $3.47467$
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  + (−2.11 + 3.65i)5-s + (2.19 + 1.47i)7-s + (0.964 + 1.67i)11-s + (−0.291 − 0.504i)13-s + (−3.61 + 6.25i)17-s + (2.10 + 3.64i)19-s + (0.639 − 1.10i)23-s + (−6.41 − 11.1i)25-s + (4.20 − 7.27i)29-s − 0.952·31-s + (−10.0 + 4.89i)35-s + (3.03 + 5.25i)37-s + (−1.31 − 2.27i)41-s + (0.442 − 0.766i)43-s − 5.76·47-s + ⋯
L(s)  = 1  + (−0.944 + 1.63i)5-s + (0.829 + 0.559i)7-s + (0.290 + 0.503i)11-s + (−0.0808 − 0.140i)13-s + (−0.875 + 1.51i)17-s + (0.482 + 0.835i)19-s + (0.133 − 0.231i)23-s + (−1.28 − 2.22i)25-s + (0.780 − 1.35i)29-s − 0.171·31-s + (−1.69 + 0.827i)35-s + (0.498 + 0.863i)37-s + (−0.205 − 0.355i)41-s + (0.0674 − 0.116i)43-s − 0.840·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.926 - 0.375i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.926 - 0.375i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $-0.926 - 0.375i$
Analytic conductor: \(12.0733\)
Root analytic conductor: \(3.47467\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1512} (1369, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1512,\ (\ :1/2),\ -0.926 - 0.375i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.150185175\)
\(L(\frac12)\) \(\approx\) \(1.150185175\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + (-2.19 - 1.47i)T \)
good5 \( 1 + (2.11 - 3.65i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.964 - 1.67i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.291 + 0.504i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.61 - 6.25i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.10 - 3.64i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.639 + 1.10i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.20 + 7.27i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 0.952T + 31T^{2} \)
37 \( 1 + (-3.03 - 5.25i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.31 + 2.27i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.442 + 0.766i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 5.76T + 47T^{2} \)
53 \( 1 + (-0.962 + 1.66i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 4.55T + 59T^{2} \)
61 \( 1 + 10.5T + 61T^{2} \)
67 \( 1 + 4.86T + 67T^{2} \)
71 \( 1 + 11.5T + 71T^{2} \)
73 \( 1 + (-0.446 + 0.772i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 11.8T + 79T^{2} \)
83 \( 1 + (-5.24 + 9.08i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (3.87 + 6.71i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.98 - 3.44i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.19537205723750419306838511525, −8.853249286355425436751352474148, −8.015921005460563497612521607819, −7.58192664591449257085070205767, −6.53472329218022279523108360307, −6.02083252743100476792172308191, −4.59475594325763999321802651699, −3.88122464230527581675966226889, −2.84825062462880426066196037956, −1.86219555277889352513352103991, 0.47870065437347648453788655530, 1.42675472474733437559322936594, 3.11931801577350480129423784096, 4.37278237577150265100885857568, 4.72176446603094062888839314019, 5.50399387844092783074389286020, 7.01609070992177792560852985558, 7.52564389306341792118471289560, 8.512075512410194867220008705505, 8.898333494132808741022224456510

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