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Properties

Label 2-1512-63.58-c1-0-21
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: $\chi_{1512} (793, \cdot )$
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

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

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