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Properties

Label 2-5-5.4-c33-0-2
Degree $2$
Conductor $5$
Sign $-0.360 - 0.932i$
Analytic cond. $34.4914$
Root an. cond. $5.87293$
Motivic weight $33$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.14e5i·2-s + 6.13e7i·3-s − 4.62e9·4-s + (1.22e11 + 3.18e11i)5-s + 7.04e12·6-s + 1.47e14i·7-s − 4.56e14i·8-s + 1.80e15·9-s + (3.65e16 − 1.41e16i)10-s − 9.00e16·11-s − 2.83e17i·12-s + 2.24e18i·13-s + 1.69e19·14-s + (−1.95e19 + 7.53e18i)15-s − 9.21e19·16-s − 1.71e20i·17-s + ⋯
L(s)  = 1  − 1.24i·2-s + 0.822i·3-s − 0.538·4-s + (0.360 + 0.932i)5-s + 1.01·6-s + 1.67i·7-s − 0.572i·8-s + 0.323·9-s + (1.15 − 0.446i)10-s − 0.590·11-s − 0.442i·12-s + 0.933i·13-s + 2.07·14-s + (−0.767 + 0.296i)15-s − 1.24·16-s − 0.852i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.360 - 0.932i)\, \overline{\Lambda}(34-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+33/2) \, L(s)\cr =\mathstrut & (-0.360 - 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $-0.360 - 0.932i$
Analytic conductor: \(34.4914\)
Root analytic conductor: \(5.87293\)
Motivic weight: \(33\)
Rational: no
Arithmetic: yes
Character: $\chi_{5} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5,\ (\ :33/2),\ -0.360 - 0.932i)\)

Particular Values

\(L(17)\) \(\approx\) \(1.285145328\)
\(L(\frac12)\) \(\approx\) \(1.285145328\)
\(L(\frac{35}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.22e11 - 3.18e11i)T \)
good2 \( 1 + 1.14e5iT - 8.58e9T^{2} \)
3 \( 1 - 6.13e7iT - 5.55e15T^{2} \)
7 \( 1 - 1.47e14iT - 7.73e27T^{2} \)
11 \( 1 + 9.00e16T + 2.32e34T^{2} \)
13 \( 1 - 2.24e18iT - 5.75e36T^{2} \)
17 \( 1 + 1.71e20iT - 4.02e40T^{2} \)
19 \( 1 + 2.31e21T + 1.58e42T^{2} \)
23 \( 1 + 4.04e22iT - 8.65e44T^{2} \)
29 \( 1 + 2.58e23T + 1.81e48T^{2} \)
31 \( 1 - 3.06e24T + 1.64e49T^{2} \)
37 \( 1 + 1.35e25iT - 5.63e51T^{2} \)
41 \( 1 - 2.64e26T + 1.66e53T^{2} \)
43 \( 1 - 1.37e27iT - 8.02e53T^{2} \)
47 \( 1 + 5.93e26iT - 1.51e55T^{2} \)
53 \( 1 + 1.16e28iT - 7.96e56T^{2} \)
59 \( 1 + 1.75e29T + 2.74e58T^{2} \)
61 \( 1 - 7.22e28T + 8.23e58T^{2} \)
67 \( 1 - 1.22e29iT - 1.82e60T^{2} \)
71 \( 1 + 3.82e30T + 1.23e61T^{2} \)
73 \( 1 - 4.64e30iT - 3.08e61T^{2} \)
79 \( 1 + 1.29e31T + 4.18e62T^{2} \)
83 \( 1 + 7.76e30iT - 2.13e63T^{2} \)
89 \( 1 + 7.56e31T + 2.13e64T^{2} \)
97 \( 1 - 4.71e32iT - 3.65e65T^{2} \)
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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

−15.95181347526245876381237918321, −14.80324976518271836461610554679, −12.73552636119821422097756322715, −11.34831485213088323980509295129, −10.23449568769311548591324217118, −9.094695510784760223492936301628, −6.42498365253522622027820990073, −4.48500292558362072961059505647, −2.80230869744480025637731965390, −2.08233193468440573083922829215, 0.35561410797146454839400122623, 1.69251293357812484104202200864, 4.39160317623528326837952280054, 5.99365794106868192859057553955, 7.32639737382751735040845084998, 8.217531935886005133306546453778, 10.44134663175954399571224160276, 12.87020651409958291609470026736, 13.68034786727700275206729218160, 15.48706077882065704543745779626

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