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Properties

Label 2-5-5.3-c34-0-15
Degree $2$
Conductor $5$
Sign $0.954 - 0.298i$
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  + (1.74e5 − 1.74e5i)2-s + (−1.58e8 − 1.58e8i)3-s − 4.35e10i·4-s + (−4.99e11 + 5.76e11i)5-s − 5.52e13·6-s + (3.74e13 − 3.74e13i)7-s + (−4.60e15 − 4.60e15i)8-s + 3.36e16i·9-s + (1.34e16 + 1.87e17i)10-s + 3.71e17·11-s + (−6.91e18 + 6.91e18i)12-s + (−5.86e18 − 5.86e18i)13-s − 1.30e19i·14-s + (1.70e20 − 1.22e19i)15-s − 8.55e20·16-s + (−5.97e20 + 5.97e20i)17-s + ⋯
L(s)  = 1  + (1.32 − 1.32i)2-s + (−1.22 − 1.22i)3-s − 2.53i·4-s + (−0.654 + 0.755i)5-s − 3.26·6-s + (0.161 − 0.161i)7-s + (−2.04 − 2.04i)8-s + 2.01i·9-s + (0.134 + 1.87i)10-s + 0.735·11-s + (−3.11 + 3.11i)12-s + (−0.677 − 0.677i)13-s − 0.428i·14-s + (1.73 − 0.124i)15-s − 2.89·16-s + (−0.722 + 0.722i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.954 - 0.298i)\, \overline{\Lambda}(35-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+17) \, L(s)\cr =\mathstrut & (0.954 - 0.298i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $0.954 - 0.298i$
Analytic conductor: \(36.6128\)
Root analytic conductor: \(6.05085\)
Motivic weight: \(34\)
Rational: no
Arithmetic: yes
Character: $\chi_{5} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5,\ (\ :17),\ 0.954 - 0.298i)\)

Particular Values

\(L(\frac{35}{2})\) \(\approx\) \(0.3521122086\)
\(L(\frac12)\) \(\approx\) \(0.3521122086\)
\(L(18)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (4.99e11 - 5.76e11i)T \)
good2 \( 1 + (-1.74e5 + 1.74e5i)T - 1.71e10iT^{2} \)
3 \( 1 + (1.58e8 + 1.58e8i)T + 1.66e16iT^{2} \)
7 \( 1 + (-3.74e13 + 3.74e13i)T - 5.41e28iT^{2} \)
11 \( 1 - 3.71e17T + 2.55e35T^{2} \)
13 \( 1 + (5.86e18 + 5.86e18i)T + 7.48e37iT^{2} \)
17 \( 1 + (5.97e20 - 5.97e20i)T - 6.84e41iT^{2} \)
19 \( 1 + 4.17e21iT - 3.00e43T^{2} \)
23 \( 1 + (-4.18e22 - 4.18e22i)T + 1.98e46iT^{2} \)
29 \( 1 + 4.95e24iT - 5.26e49T^{2} \)
31 \( 1 + 8.54e24T + 5.08e50T^{2} \)
37 \( 1 + (2.77e26 - 2.77e26i)T - 2.08e53iT^{2} \)
41 \( 1 + 3.86e26T + 6.83e54T^{2} \)
43 \( 1 + (-4.26e27 - 4.26e27i)T + 3.45e55iT^{2} \)
47 \( 1 + (-1.94e28 + 1.94e28i)T - 7.10e56iT^{2} \)
53 \( 1 + (-7.39e28 - 7.39e28i)T + 4.22e58iT^{2} \)
59 \( 1 - 2.05e30iT - 1.61e60T^{2} \)
61 \( 1 - 6.46e29T + 5.02e60T^{2} \)
67 \( 1 + (4.39e30 - 4.39e30i)T - 1.22e62iT^{2} \)
71 \( 1 + 1.23e31T + 8.76e62T^{2} \)
73 \( 1 + (6.30e31 + 6.30e31i)T + 2.25e63iT^{2} \)
79 \( 1 - 2.99e32iT - 3.30e64T^{2} \)
83 \( 1 + (5.28e32 + 5.28e32i)T + 1.77e65iT^{2} \)
89 \( 1 - 4.19e32iT - 1.90e66T^{2} \)
97 \( 1 + (7.80e33 - 7.80e33i)T - 3.55e67iT^{2} \)
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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

−13.52466074381805619039144185281, −12.35022459210111557039742043112, −11.48472499192443942353410997803, −10.60700424616875444388349589238, −7.04628238227731383938386123144, −5.84418080531829167957027522261, −4.35166600382852421866579288288, −2.62698102910728396886206494145, −1.27703404315155639859204780392, −0.081936734066021083320245236943, 3.82535015704171213829027170375, 4.62491037493546451186910825240, 5.50855361391277082112263155701, 6.96259869193559067815436928628, 8.981869906121242446263005154674, 11.53071001686804754136919451618, 12.40911292682125711839094484996, 14.50080891883189223520925897378, 15.71633287238005157463493694727, 16.48284049844662891642195886650

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