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Properties

Label 2-1260-7.2-c1-0-3
Degree $2$
Conductor $1260$
Sign $0.266 - 0.963i$
Analytic cond. $10.0611$
Root an. cond. $3.17193$
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  + (−0.5 + 0.866i)5-s + (2.5 + 0.866i)7-s + (3 + 5.19i)11-s + 2·13-s + (−3 − 5.19i)17-s + (−4 + 6.92i)19-s + (1.5 − 2.59i)23-s + (−0.499 − 0.866i)25-s − 3·29-s + (−1 − 1.73i)31-s + (−2 + 1.73i)35-s + (−4 + 6.92i)37-s + 3·41-s + 5·43-s + (5.5 + 4.33i)49-s + ⋯
L(s)  = 1  + (−0.223 + 0.387i)5-s + (0.944 + 0.327i)7-s + (0.904 + 1.56i)11-s + 0.554·13-s + (−0.727 − 1.26i)17-s + (−0.917 + 1.58i)19-s + (0.312 − 0.541i)23-s + (−0.0999 − 0.173i)25-s − 0.557·29-s + (−0.179 − 0.311i)31-s + (−0.338 + 0.292i)35-s + (−0.657 + 1.13i)37-s + 0.468·41-s + 0.762·43-s + (0.785 + 0.618i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.266 - 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.266 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1260\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.266 - 0.963i$
Analytic conductor: \(10.0611\)
Root analytic conductor: \(3.17193\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1260} (541, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1260,\ (\ :1/2),\ 0.266 - 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.699456644\)
\(L(\frac12)\) \(\approx\) \(1.699456644\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (-2.5 - 0.866i)T \)
good11 \( 1 + (-3 - 5.19i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + (3 + 5.19i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4 - 6.92i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.5 + 2.59i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 3T + 29T^{2} \)
31 \( 1 + (1 + 1.73i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4 - 6.92i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 3T + 41T^{2} \)
43 \( 1 - 5T + 43T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6 - 10.3i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.5 - 6.06i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (-5 - 8.66i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2 + 3.46i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 3T + 83T^{2} \)
89 \( 1 + (1.5 - 2.59i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 10T + 97T^{2} \)
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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

−9.830876611086820068999627703096, −9.024932152330816997235176021680, −8.245006239745881421443089921481, −7.34278921995405715382349004648, −6.69817117633524420057865160748, −5.67457652762456725804046782168, −4.54780852555171622326869929089, −4.01258262723753373635712756109, −2.48677459122370446479614714613, −1.54599485178362962683927342140, 0.76542102191488171360569893374, 1.98989712732978514722706053451, 3.59137872647668111989486810239, 4.21278949124244672537288753854, 5.29080510672150683254101321524, 6.18881934609651796150794330490, 7.02161594270452906245617475796, 8.120293836222662568042044607843, 8.754252631221316689724545015977, 9.126428047352636703313038229226

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