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Properties

Label 2-468-117.110-c1-0-10
Degree $2$
Conductor $468$
Sign $0.379 + 0.925i$
Analytic cond. $3.73699$
Root an. cond. $1.93313$
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  + (0.706 − 1.58i)3-s + (0.956 − 3.56i)5-s + (3.66 + 3.66i)7-s + (−2.00 − 2.23i)9-s + (3.54 + 0.949i)11-s + (1.38 + 3.32i)13-s + (−4.96 − 4.03i)15-s + (−1.54 − 2.67i)17-s + (0.0965 + 0.0258i)19-s + (8.38 − 3.20i)21-s − 4.53·23-s + (−7.49 − 4.32i)25-s + (−4.94 + 1.59i)27-s + (−5.03 + 2.90i)29-s + (−4.10 − 1.10i)31-s + ⋯
L(s)  = 1  + (0.407 − 0.913i)3-s + (0.427 − 1.59i)5-s + (1.38 + 1.38i)7-s + (−0.667 − 0.744i)9-s + (1.06 + 0.286i)11-s + (0.384 + 0.923i)13-s + (−1.28 − 1.04i)15-s + (−0.374 − 0.649i)17-s + (0.0221 + 0.00593i)19-s + (1.83 − 0.700i)21-s − 0.944·23-s + (−1.49 − 0.865i)25-s + (−0.951 + 0.306i)27-s + (−0.935 + 0.540i)29-s + (−0.737 − 0.197i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 468 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.379 + 0.925i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 468 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.379 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(468\)    =    \(2^{2} \cdot 3^{2} \cdot 13\)
Sign: $0.379 + 0.925i$
Analytic conductor: \(3.73699\)
Root analytic conductor: \(1.93313\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{468} (461, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 468,\ (\ :1/2),\ 0.379 + 0.925i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.62453 - 1.08986i\)
\(L(\frac12)\) \(\approx\) \(1.62453 - 1.08986i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.706 + 1.58i)T \)
13 \( 1 + (-1.38 - 3.32i)T \)
good5 \( 1 + (-0.956 + 3.56i)T + (-4.33 - 2.5i)T^{2} \)
7 \( 1 + (-3.66 - 3.66i)T + 7iT^{2} \)
11 \( 1 + (-3.54 - 0.949i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (1.54 + 2.67i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.0965 - 0.0258i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + 4.53T + 23T^{2} \)
29 \( 1 + (5.03 - 2.90i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.10 + 1.10i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (4.69 - 1.25i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-0.971 - 0.971i)T + 41iT^{2} \)
43 \( 1 + 8.59iT - 43T^{2} \)
47 \( 1 + (-0.442 - 1.64i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 - 5.85iT - 53T^{2} \)
59 \( 1 + (-2.54 - 9.51i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 - 6.74T + 61T^{2} \)
67 \( 1 + (3.02 - 3.02i)T - 67iT^{2} \)
71 \( 1 + (-2.87 + 10.7i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (1.26 + 1.26i)T + 73iT^{2} \)
79 \( 1 + (1.68 - 2.91i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.28 + 0.344i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (3.45 + 12.9i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (1.70 - 1.70i)T - 97iT^{2} \)
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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

−11.33505589060818083465447991599, −9.348064410717160264023882851863, −8.928604681094170974854001779650, −8.468333490096490284716655315747, −7.34078177584297309425443624939, −6.02608930574229060396927135077, −5.26237296202783041332432214086, −4.18165152051032976752114345068, −2.05184601618049653260199922124, −1.50674522210321958089480213148, 1.93668731698029900354984878757, 3.51903965631018841336890659996, 4.06942254387154420688595770978, 5.49791811186534098377919603170, 6.62848003521104606431246763815, 7.65475992479255517544822702558, 8.383310419937108841051665607998, 9.719900102445991433332709317601, 10.43751882301952296197871445050, 11.01534150278490619830895351865

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