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Properties

Label 2-504-63.38-c1-0-4
Degree $2$
Conductor $504$
Sign $-0.965 + 0.258i$
Analytic cond. $4.02446$
Root an. cond. $2.00610$
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  + (−1.41 + 0.991i)3-s + (−1.11 + 1.92i)5-s + (0.362 + 2.62i)7-s + (1.03 − 2.81i)9-s + (−1.01 + 0.586i)11-s + (−3.12 + 1.80i)13-s + (−0.331 − 3.83i)15-s + (3.71 − 6.42i)17-s + (−3.05 + 1.76i)19-s + (−3.11 − 3.36i)21-s + (−5.03 − 2.90i)23-s + (0.0337 + 0.0583i)25-s + (1.32 + 5.02i)27-s + (−6.04 − 3.48i)29-s + 7.95i·31-s + ⋯
L(s)  = 1  + (−0.819 + 0.572i)3-s + (−0.496 + 0.860i)5-s + (0.136 + 0.990i)7-s + (0.344 − 0.938i)9-s + (−0.306 + 0.176i)11-s + (−0.866 + 0.500i)13-s + (−0.0854 − 0.989i)15-s + (0.900 − 1.55i)17-s + (−0.700 + 0.404i)19-s + (−0.679 − 0.733i)21-s + (−1.04 − 0.605i)23-s + (0.00674 + 0.0116i)25-s + (0.255 + 0.966i)27-s + (−1.12 − 0.648i)29-s + 1.42i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 + 0.258i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.965 + 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.965 + 0.258i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ -0.965 + 0.258i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0512752 - 0.389208i\)
\(L(\frac12)\) \(\approx\) \(0.0512752 - 0.389208i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.41 - 0.991i)T \)
7 \( 1 + (-0.362 - 2.62i)T \)
good5 \( 1 + (1.11 - 1.92i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.01 - 0.586i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.12 - 1.80i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.71 + 6.42i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.05 - 1.76i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (5.03 + 2.90i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (6.04 + 3.48i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 7.95iT - 31T^{2} \)
37 \( 1 + (5.54 + 9.60i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.809 - 1.40i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.904 + 1.56i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 8.52T + 47T^{2} \)
53 \( 1 + (-9.62 - 5.55i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + 4.00T + 59T^{2} \)
61 \( 1 - 8.18iT - 61T^{2} \)
67 \( 1 - 9.92T + 67T^{2} \)
71 \( 1 - 3.67iT - 71T^{2} \)
73 \( 1 + (-6.92 - 3.99i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + 4.51T + 79T^{2} \)
83 \( 1 + (-0.390 + 0.677i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (1.75 + 3.03i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3.49 - 2.01i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.47907102564109765728041364121, −10.54392609690970209375443862453, −9.792118300016561274157851954611, −8.928096938604685314846550852511, −7.59339387681327870444132687387, −6.82549405194613873974222925851, −5.70285928529325936482408111862, −4.91884363869074627918578950004, −3.68532244449733039673481405920, −2.41618736294908365488516080343, 0.25342730393664877555357023022, 1.70329776522526929441333534325, 3.77584742977973320943112554736, 4.78924644505109242873787972557, 5.68633134125071693423970680629, 6.79324257573727326280582999178, 7.926028151266858949809073399011, 8.137175637278374949795956546738, 9.823783149680721910669047828300, 10.51199333753332990418412270767

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