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Properties

Label 2-5-5.4-c33-0-3
Degree $2$
Conductor $5$
Sign $0.843 + 0.537i$
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  − 1.56e5i·2-s + 1.41e8i·3-s − 1.59e10·4-s + (−2.87e11 − 1.83e11i)5-s + 2.21e13·6-s − 8.25e13i·7-s + 1.14e15i·8-s − 1.44e16·9-s + (−2.87e16 + 4.50e16i)10-s − 6.60e16·11-s − 2.24e18i·12-s + 2.09e18i·13-s − 1.29e19·14-s + (2.59e19 − 4.06e19i)15-s + 4.26e19·16-s + 1.67e18i·17-s + ⋯
L(s)  = 1  − 1.68i·2-s + 1.89i·3-s − 1.85·4-s + (−0.843 − 0.537i)5-s + 3.20·6-s − 0.938i·7-s + 1.43i·8-s − 2.59·9-s + (−0.907 + 1.42i)10-s − 0.433·11-s − 3.51i·12-s + 0.874i·13-s − 1.58·14-s + (1.01 − 1.59i)15-s + 0.578·16-s + 0.00836i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.843 + 0.537i)\, \overline{\Lambda}(34-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+33/2) \, L(s)\cr =\mathstrut & (0.843 + 0.537i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $0.843 + 0.537i$
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: \((2,\ 5,\ (\ :33/2),\ 0.843 + 0.537i)\)

Particular Values

\(L(17)\) \(\approx\) \(1.038560792\)
\(L(\frac12)\) \(\approx\) \(1.038560792\)
\(L(\frac{35}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2.87e11 + 1.83e11i)T \)
good2 \( 1 + 1.56e5iT - 8.58e9T^{2} \)
3 \( 1 - 1.41e8iT - 5.55e15T^{2} \)
7 \( 1 + 8.25e13iT - 7.73e27T^{2} \)
11 \( 1 + 6.60e16T + 2.32e34T^{2} \)
13 \( 1 - 2.09e18iT - 5.75e36T^{2} \)
17 \( 1 - 1.67e18iT - 4.02e40T^{2} \)
19 \( 1 - 3.21e20T + 1.58e42T^{2} \)
23 \( 1 + 1.95e22iT - 8.65e44T^{2} \)
29 \( 1 - 6.09e23T + 1.81e48T^{2} \)
31 \( 1 - 7.68e24T + 1.64e49T^{2} \)
37 \( 1 - 7.12e25iT - 5.63e51T^{2} \)
41 \( 1 - 1.56e26T + 1.66e53T^{2} \)
43 \( 1 + 1.27e27iT - 8.02e53T^{2} \)
47 \( 1 - 5.00e27iT - 1.51e55T^{2} \)
53 \( 1 - 2.77e28iT - 7.96e56T^{2} \)
59 \( 1 - 1.46e28T + 2.74e58T^{2} \)
61 \( 1 - 2.37e29T + 8.23e58T^{2} \)
67 \( 1 + 4.57e29iT - 1.82e60T^{2} \)
71 \( 1 + 2.86e30T + 1.23e61T^{2} \)
73 \( 1 + 1.51e30iT - 3.08e61T^{2} \)
79 \( 1 + 1.25e30T + 4.18e62T^{2} \)
83 \( 1 - 1.16e31iT - 2.13e63T^{2} \)
89 \( 1 + 1.11e32T + 2.13e64T^{2} \)
97 \( 1 + 4.59e32iT - 3.65e65T^{2} \)
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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

−15.72353810431194707149253689998, −13.91204525785493116726112255375, −11.90234834271866234694264150991, −10.83768467721235899150241289755, −9.898657265505565760727802093038, −8.627965599642129850021183674242, −4.66140047627259005194734013631, −4.14481327962343609488644277696, −2.96633360103242603796700933981, −0.69527156645341607379172660968, 0.53631729124311603658077775512, 2.69751071204199871821443326000, 5.51075593065049130451507106228, 6.60511454142806997603079846540, 7.70619273449086288541070462416, 8.406167459102682394503631026672, 11.81924150402798834719526495105, 13.18198721097925605962800986628, 14.54797831598055145571088550025, 15.73496495808674943513864562968

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