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Properties

Label 2-1260-7.4-c1-0-5
Degree $2$
Conductor $1260$
Sign $0.386 - 0.922i$
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 + (−1 + 1.73i)11-s + 13-s + (−2 + 3.46i)17-s + (0.5 + 0.866i)19-s + (2 + 3.46i)23-s + (−0.499 + 0.866i)25-s + (2.5 − 4.33i)31-s + (0.500 + 2.59i)35-s + (2.5 + 4.33i)37-s − 2·41-s − 9·43-s + (−1 − 1.73i)47-s + (5.5 + 4.33i)49-s + ⋯
L(s)  = 1  + (0.223 + 0.387i)5-s + (0.944 + 0.327i)7-s + (−0.301 + 0.522i)11-s + 0.277·13-s + (−0.485 + 0.840i)17-s + (0.114 + 0.198i)19-s + (0.417 + 0.722i)23-s + (−0.0999 + 0.173i)25-s + (0.449 − 0.777i)31-s + (0.0845 + 0.439i)35-s + (0.410 + 0.711i)37-s − 0.312·41-s − 1.37·43-s + (−0.145 − 0.252i)47-s + (0.785 + 0.618i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1260\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.386 - 0.922i$
Analytic conductor: \(10.0611\)
Root analytic conductor: \(3.17193\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1260} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1260,\ (\ :1/2),\ 0.386 - 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.781465661\)
\(L(\frac12)\) \(\approx\) \(1.781465661\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 - 1.73i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - T + 13T^{2} \)
17 \( 1 + (2 - 3.46i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2 - 3.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + (-2.5 + 4.33i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.5 - 4.33i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 + 9T + 43T^{2} \)
47 \( 1 + (1 + 1.73i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6 + 10.3i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4 - 6.92i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7 - 12.1i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.5 - 7.79i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 2T + 71T^{2} \)
73 \( 1 + (0.5 - 0.866i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.5 - 2.59i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 18T + 83T^{2} \)
89 \( 1 + (2 + 3.46i)T + (-44.5 + 77.0i)T^{2} \)
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

−9.990444410354647229302745913246, −8.928663796040093836047698623243, −8.230579958708695459374799381180, −7.45350287415078392002642695054, −6.52610152503615364415170469310, −5.61838532994016418054795870353, −4.79041846927210772705929051001, −3.77647361642209625806480189662, −2.49943537853266252991906759631, −1.52587005309749566694790347880, 0.795051122441443070129289459852, 2.11069823569327479181775525020, 3.33335391587404270598680083218, 4.62466264764438828039542901196, 5.09304651371527457379409601972, 6.20321124691495288460081706449, 7.11736147657663508784286328952, 8.032738067929047079923022482024, 8.673426588443494462754777651607, 9.424569012560558498972933647611

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