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Properties

Label 2-115-5.4-c5-0-12
Degree $2$
Conductor $115$
Sign $-0.171 + 0.985i$
Analytic cond. $18.4441$
Root an. cond. $4.29466$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 8.29i·2-s + 20.3i·3-s − 36.8·4-s + (55.0 + 9.59i)5-s − 168.·6-s + 101. i·7-s − 39.8i·8-s − 169.·9-s + (−79.5 + 456. i)10-s − 362.·11-s − 747. i·12-s − 540. i·13-s − 845.·14-s + (−194. + 1.11e3i)15-s − 847.·16-s + 561. i·17-s + ⋯
L(s)  = 1  + 1.46i·2-s + 1.30i·3-s − 1.15·4-s + (0.985 + 0.171i)5-s − 1.91·6-s + 0.786i·7-s − 0.219i·8-s − 0.698·9-s + (−0.251 + 1.44i)10-s − 0.903·11-s − 1.49i·12-s − 0.886i·13-s − 1.15·14-s + (−0.223 + 1.28i)15-s − 0.827·16-s + 0.471i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.171 + 0.985i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.171 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $-0.171 + 0.985i$
Analytic conductor: \(18.4441\)
Root analytic conductor: \(4.29466\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{115} (24, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 115,\ (\ :5/2),\ -0.171 + 0.985i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.724851075\)
\(L(\frac12)\) \(\approx\) \(1.724851075\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-55.0 - 9.59i)T \)
23 \( 1 + 529iT \)
good2 \( 1 - 8.29iT - 32T^{2} \)
3 \( 1 - 20.3iT - 243T^{2} \)
7 \( 1 - 101. iT - 1.68e4T^{2} \)
11 \( 1 + 362.T + 1.61e5T^{2} \)
13 \( 1 + 540. iT - 3.71e5T^{2} \)
17 \( 1 - 561. iT - 1.41e6T^{2} \)
19 \( 1 + 1.11e3T + 2.47e6T^{2} \)
29 \( 1 - 7.26e3T + 2.05e7T^{2} \)
31 \( 1 - 450.T + 2.86e7T^{2} \)
37 \( 1 - 4.96e3iT - 6.93e7T^{2} \)
41 \( 1 - 7.55e3T + 1.15e8T^{2} \)
43 \( 1 - 1.60e4iT - 1.47e8T^{2} \)
47 \( 1 + 1.94e4iT - 2.29e8T^{2} \)
53 \( 1 - 5.35e3iT - 4.18e8T^{2} \)
59 \( 1 + 4.93e4T + 7.14e8T^{2} \)
61 \( 1 - 1.63e3T + 8.44e8T^{2} \)
67 \( 1 + 3.44e4iT - 1.35e9T^{2} \)
71 \( 1 + 5.39e3T + 1.80e9T^{2} \)
73 \( 1 - 5.24e4iT - 2.07e9T^{2} \)
79 \( 1 + 4.43e3T + 3.07e9T^{2} \)
83 \( 1 - 9.18e4iT - 3.93e9T^{2} \)
89 \( 1 - 8.87e4T + 5.58e9T^{2} \)
97 \( 1 - 5.56e4iT - 8.58e9T^{2} \)
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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

−13.78967302429692143960492154024, −12.67172748455404224041258252228, −10.81879547056388132248357173632, −10.04942344640460667265339055995, −8.964950815110959858449691136681, −8.074439358485842620303104597548, −6.42108048482741060673798472673, −5.52656829008007783805803883415, −4.72517758308867770575037365780, −2.70189249286172917619152542473, 0.62931305016555509234678932482, 1.71914438263393115023555760907, 2.66246861983355665848731167472, 4.53480990403258999517752986462, 6.32691510820317620921661145660, 7.38218327996330290651196602103, 8.896503689572419184503153731089, 10.08197249101645712285869536207, 10.85161134779635722526557843428, 12.09142181119958421857956983013

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