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Properties

Label 2-5-5.2-c34-0-2
Degree $2$
Conductor $5$
Sign $0.328 - 0.944i$
Analytic cond. $36.6128$
Root an. cond. $6.05085$
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  + (−8.52e4 − 8.52e4i)2-s + (−1.68e8 + 1.68e8i)3-s − 2.62e9i·4-s + (−5.92e11 + 4.80e11i)5-s + 2.86e13·6-s + (2.04e14 + 2.04e14i)7-s + (−1.68e15 + 1.68e15i)8-s − 3.98e16i·9-s + (9.15e16 + 9.49e15i)10-s − 4.22e17·11-s + (4.42e17 + 4.42e17i)12-s + (8.97e18 − 8.97e18i)13-s − 3.49e19i·14-s + (1.87e19 − 1.80e20i)15-s + 2.43e20·16-s + (−2.34e20 − 2.34e20i)17-s + ⋯
L(s)  = 1  + (−0.650 − 0.650i)2-s + (−1.30 + 1.30i)3-s − 0.153i·4-s + (−0.776 + 0.630i)5-s + 1.69·6-s + (0.881 + 0.881i)7-s + (−0.750 + 0.750i)8-s − 2.38i·9-s + (0.915 + 0.0949i)10-s − 0.836·11-s + (0.199 + 0.199i)12-s + (1.03 − 1.03i)13-s − 1.14i·14-s + (0.190 − 1.83i)15-s + 0.823·16-s + (−0.283 − 0.283i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.328 - 0.944i)\, \overline{\Lambda}(35-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+17) \, L(s)\cr =\mathstrut & (0.328 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $0.328 - 0.944i$
Analytic conductor: \(36.6128\)
Root analytic conductor: \(6.05085\)
Motivic weight: \(34\)
Rational: no
Arithmetic: yes
Character: $\chi_{5} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5,\ (\ :17),\ 0.328 - 0.944i)\)

Particular Values

\(L(\frac{35}{2})\) \(\approx\) \(0.5304805536\)
\(L(\frac12)\) \(\approx\) \(0.5304805536\)
\(L(18)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (5.92e11 - 4.80e11i)T \)
good2 \( 1 + (8.52e4 + 8.52e4i)T + 1.71e10iT^{2} \)
3 \( 1 + (1.68e8 - 1.68e8i)T - 1.66e16iT^{2} \)
7 \( 1 + (-2.04e14 - 2.04e14i)T + 5.41e28iT^{2} \)
11 \( 1 + 4.22e17T + 2.55e35T^{2} \)
13 \( 1 + (-8.97e18 + 8.97e18i)T - 7.48e37iT^{2} \)
17 \( 1 + (2.34e20 + 2.34e20i)T + 6.84e41iT^{2} \)
19 \( 1 + 3.96e21iT - 3.00e43T^{2} \)
23 \( 1 + (1.87e22 - 1.87e22i)T - 1.98e46iT^{2} \)
29 \( 1 - 3.22e24iT - 5.26e49T^{2} \)
31 \( 1 - 3.15e25T + 5.08e50T^{2} \)
37 \( 1 + (-3.27e26 - 3.27e26i)T + 2.08e53iT^{2} \)
41 \( 1 + 9.81e26T + 6.83e54T^{2} \)
43 \( 1 + (-2.79e27 + 2.79e27i)T - 3.45e55iT^{2} \)
47 \( 1 + (9.12e27 + 9.12e27i)T + 7.10e56iT^{2} \)
53 \( 1 + (-2.57e28 + 2.57e28i)T - 4.22e58iT^{2} \)
59 \( 1 - 7.57e29iT - 1.61e60T^{2} \)
61 \( 1 + 3.27e30T + 5.02e60T^{2} \)
67 \( 1 + (-3.29e30 - 3.29e30i)T + 1.22e62iT^{2} \)
71 \( 1 + 1.48e31T + 8.76e62T^{2} \)
73 \( 1 + (4.68e31 - 4.68e31i)T - 2.25e63iT^{2} \)
79 \( 1 - 5.17e31iT - 3.30e64T^{2} \)
83 \( 1 + (-1.93e32 + 1.93e32i)T - 1.77e65iT^{2} \)
89 \( 1 + 4.72e32iT - 1.90e66T^{2} \)
97 \( 1 + (5.55e33 + 5.55e33i)T + 3.55e67iT^{2} \)
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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

−15.67021597024722205534574295666, −15.14103098227730897598879942020, −11.79293095047819705559559871043, −11.06401229500127546240355734125, −10.24175820769218762442650875554, −8.543027190863621560488956310285, −5.91197857933452556735380119821, −4.80800212227049288980970676815, −2.94143535497844674622542976530, −0.71405370903729257502039509165, 0.41142575932862499927767371565, 1.40932896552020044820431706932, 4.42268388211510973666971214045, 6.21608513230326161641542232969, 7.52090781515947437414705861176, 8.211945547657710560972552637239, 11.02474744727165958270262225030, 12.13249354464516918879604860731, 13.40445013206841900161751666091, 16.07947003754579783354395811093

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