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Properties

Label 2-2e2-4.3-c34-0-6
Degree $2$
Conductor $4$
Sign $0.716 - 0.697i$
Analytic cond. $29.2902$
Root an. cond. $5.41204$
Motivic weight $34$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4.93e4 + 1.21e5i)2-s − 9.14e6i·3-s + (−1.23e10 + 1.19e10i)4-s − 6.20e11·5-s + (1.11e12 − 4.51e11i)6-s − 3.03e14i·7-s + (−2.06e15 − 9.02e14i)8-s + 1.65e16·9-s + (−3.06e16 − 7.53e16i)10-s + 4.34e17i·11-s + (1.09e17 + 1.12e17i)12-s − 1.62e18·13-s + (3.69e19 − 1.50e19i)14-s + 5.67e18i·15-s + (7.75e18 − 2.95e20i)16-s + 1.14e21·17-s + ⋯
L(s)  = 1  + (0.376 + 0.926i)2-s − 0.0708i·3-s + (−0.716 + 0.697i)4-s − 0.813·5-s + (0.0655 − 0.0266i)6-s − 1.30i·7-s + (−0.916 − 0.400i)8-s + 0.994·9-s + (−0.306 − 0.753i)10-s + 0.858i·11-s + (0.0494 + 0.0507i)12-s − 0.188·13-s + (1.21 − 0.492i)14-s + 0.0576i·15-s + (0.0262 − 0.999i)16-s + 1.38·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.716 - 0.697i)\, \overline{\Lambda}(35-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+17) \, L(s)\cr =\mathstrut & (0.716 - 0.697i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4\)    =    \(2^{2}\)
Sign: $0.716 - 0.697i$
Analytic conductor: \(29.2902\)
Root analytic conductor: \(5.41204\)
Motivic weight: \(34\)
Rational: no
Arithmetic: yes
Character: $\chi_{4} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4,\ (\ :17),\ 0.716 - 0.697i)\)

Particular Values

\(L(\frac{35}{2})\) \(\approx\) \(1.954843311\)
\(L(\frac12)\) \(\approx\) \(1.954843311\)
\(L(18)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-4.93e4 - 1.21e5i)T \)
good3 \( 1 + 9.14e6iT - 1.66e16T^{2} \)
5 \( 1 + 6.20e11T + 5.82e23T^{2} \)
7 \( 1 + 3.03e14iT - 5.41e28T^{2} \)
11 \( 1 - 4.34e17iT - 2.55e35T^{2} \)
13 \( 1 + 1.62e18T + 7.48e37T^{2} \)
17 \( 1 - 1.14e21T + 6.84e41T^{2} \)
19 \( 1 + 4.29e21iT - 3.00e43T^{2} \)
23 \( 1 - 1.36e23iT - 1.98e46T^{2} \)
29 \( 1 - 1.10e25T + 5.26e49T^{2} \)
31 \( 1 - 1.99e25iT - 5.08e50T^{2} \)
37 \( 1 - 5.08e26T + 2.08e53T^{2} \)
41 \( 1 - 5.95e26T + 6.83e54T^{2} \)
43 \( 1 - 2.03e26iT - 3.45e55T^{2} \)
47 \( 1 + 3.56e28iT - 7.10e56T^{2} \)
53 \( 1 - 2.36e29T + 4.22e58T^{2} \)
59 \( 1 - 3.03e29iT - 1.61e60T^{2} \)
61 \( 1 - 2.73e30T + 5.02e60T^{2} \)
67 \( 1 + 1.85e31iT - 1.22e62T^{2} \)
71 \( 1 + 5.00e31iT - 8.76e62T^{2} \)
73 \( 1 + 1.57e31T + 2.25e63T^{2} \)
79 \( 1 + 2.44e32iT - 3.30e64T^{2} \)
83 \( 1 - 6.99e32iT - 1.77e65T^{2} \)
89 \( 1 + 3.74e32T + 1.90e66T^{2} \)
97 \( 1 + 1.01e34T + 3.55e67T^{2} \)
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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

−16.41667481245126759784264743368, −15.19405385692430751778488524983, −13.64725924752119037858545586472, −12.21562452652884348106282349638, −9.959077172743278355992154767060, −7.73888610931352501162683783149, −7.00330378298158182904141121544, −4.72217330019361932848599524143, −3.66919797895783786702672813503, −0.804993658995093528904837607940, 0.941728456103529836444797527310, 2.69239635388059255134173640007, 4.12601581763137219113777579399, 5.76246901212470145292482091270, 8.294391715391899300475785895373, 9.962994270955096510382190533254, 11.68345608327363627313958264776, 12.62685353777512929230197240853, 14.57562069684731493275530723846, 15.94923509017099089405589217284

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