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Properties

Label 2-5-5.2-c16-0-4
Degree $2$
Conductor $5$
Sign $0.460 - 0.887i$
Analytic cond. $8.11622$
Root an. cond. $2.84889$
Motivic weight $16$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (336. + 336. i)2-s + (6.05e3 − 6.05e3i)3-s + 1.60e5i·4-s + (3.35e5 − 2.00e5i)5-s + 4.07e6·6-s + (−3.71e5 − 3.71e5i)7-s + (−3.19e7 + 3.19e7i)8-s − 3.01e7i·9-s + (1.80e8 + 4.53e7i)10-s − 2.87e8·11-s + (9.72e8 + 9.72e8i)12-s + (−9.55e7 + 9.55e7i)13-s − 2.49e8i·14-s + (8.16e8 − 3.24e9i)15-s − 1.09e10·16-s + (−6.35e8 − 6.35e8i)17-s + ⋯
L(s)  = 1  + (1.31 + 1.31i)2-s + (0.922 − 0.922i)3-s + 2.45i·4-s + (0.858 − 0.513i)5-s + 2.42·6-s + (−0.0644 − 0.0644i)7-s + (−1.90 + 1.90i)8-s − 0.701i·9-s + (1.80 + 0.453i)10-s − 1.34·11-s + (2.26 + 2.26i)12-s + (−0.117 + 0.117i)13-s − 0.169i·14-s + (0.318 − 1.26i)15-s − 2.55·16-s + (−0.0910 − 0.0910i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.460 - 0.887i)\, \overline{\Lambda}(17-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+8) \, L(s)\cr =\mathstrut & (0.460 - 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $0.460 - 0.887i$
Analytic conductor: \(8.11622\)
Root analytic conductor: \(2.84889\)
Motivic weight: \(16\)
Rational: no
Arithmetic: yes
Character: $\chi_{5} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5,\ (\ :8),\ 0.460 - 0.887i)\)

Particular Values

\(L(\frac{17}{2})\) \(\approx\) \(3.64627 + 2.21618i\)
\(L(\frac12)\) \(\approx\) \(3.64627 + 2.21618i\)
\(L(9)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-3.35e5 + 2.00e5i)T \)
good2 \( 1 + (-336. - 336. i)T + 6.55e4iT^{2} \)
3 \( 1 + (-6.05e3 + 6.05e3i)T - 4.30e7iT^{2} \)
7 \( 1 + (3.71e5 + 3.71e5i)T + 3.32e13iT^{2} \)
11 \( 1 + 2.87e8T + 4.59e16T^{2} \)
13 \( 1 + (9.55e7 - 9.55e7i)T - 6.65e17iT^{2} \)
17 \( 1 + (6.35e8 + 6.35e8i)T + 4.86e19iT^{2} \)
19 \( 1 + 1.35e10iT - 2.88e20T^{2} \)
23 \( 1 + (-5.40e10 + 5.40e10i)T - 6.13e21iT^{2} \)
29 \( 1 - 1.69e11iT - 2.50e23T^{2} \)
31 \( 1 + 1.28e12T + 7.27e23T^{2} \)
37 \( 1 + (-2.40e12 - 2.40e12i)T + 1.23e25iT^{2} \)
41 \( 1 + 5.05e12T + 6.37e25T^{2} \)
43 \( 1 + (8.53e12 - 8.53e12i)T - 1.36e26iT^{2} \)
47 \( 1 + (-6.90e12 - 6.90e12i)T + 5.66e26iT^{2} \)
53 \( 1 + (-3.19e13 + 3.19e13i)T - 3.87e27iT^{2} \)
59 \( 1 - 2.15e14iT - 2.15e28T^{2} \)
61 \( 1 - 2.85e14T + 3.67e28T^{2} \)
67 \( 1 + (-3.15e14 - 3.15e14i)T + 1.64e29iT^{2} \)
71 \( 1 - 3.15e14T + 4.16e29T^{2} \)
73 \( 1 + (3.46e14 - 3.46e14i)T - 6.50e29iT^{2} \)
79 \( 1 + 1.88e15iT - 2.30e30T^{2} \)
83 \( 1 + (2.43e14 - 2.43e14i)T - 5.07e30iT^{2} \)
89 \( 1 - 4.86e15iT - 1.54e31T^{2} \)
97 \( 1 + (5.87e15 + 5.87e15i)T + 6.14e31iT^{2} \)
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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

−20.50164810388739868924077451289, −18.11897246411084542695540820685, −16.46071073853411604617790571496, −14.75644610556727832973120901676, −13.40358124355720367142637661414, −12.90979453525671019431185098031, −8.463962191401745890598280330674, −6.99902259455931903831199864844, −5.16338308689794282022283593666, −2.62777263027109366004750586380, 2.26489030504391811823621079441, 3.47661038553769700523810931346, 5.37071411338205940400230863004, 9.676758196297768494111669930272, 10.73620235004847716049125647864, 13.02256109328682304328807414980, 14.25097296435843916377344366114, 15.32984912306655155840872136510, 18.64237833446538640810836437485, 20.27414728122940404473024578575

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