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Properties

Label 2-115-5.4-c1-0-7
Degree $2$
Conductor $115$
Sign $0.377 + 0.925i$
Analytic cond. $0.918279$
Root an. cond. $0.958269$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.291i·2-s − 3.14i·3-s + 1.91·4-s + (−2.07 + 0.844i)5-s + 0.915·6-s − 1.20i·7-s + 1.14i·8-s − 6.86·9-s + (−0.246 − 0.603i)10-s + 3.65·11-s − 6.01i·12-s − 0.859i·13-s + 0.351·14-s + (2.65 + 6.50i)15-s + 3.49·16-s + 6.72i·17-s + ⋯
L(s)  = 1  + 0.206i·2-s − 1.81i·3-s + 0.957·4-s + (−0.925 + 0.377i)5-s + 0.373·6-s − 0.456i·7-s + 0.403i·8-s − 2.28·9-s + (−0.0778 − 0.190i)10-s + 1.10·11-s − 1.73i·12-s − 0.238i·13-s + 0.0939·14-s + (0.685 + 1.67i)15-s + 0.874·16-s + 1.63i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.377 + 0.925i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.377 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $0.377 + 0.925i$
Analytic conductor: \(0.918279\)
Root analytic conductor: \(0.958269\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{115} (24, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 115,\ (\ :1/2),\ 0.377 + 0.925i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.920217 - 0.618405i\)
\(L(\frac12)\) \(\approx\) \(0.920217 - 0.618405i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2.07 - 0.844i)T \)
23 \( 1 - iT \)
good2 \( 1 - 0.291iT - 2T^{2} \)
3 \( 1 + 3.14iT - 3T^{2} \)
7 \( 1 + 1.20iT - 7T^{2} \)
11 \( 1 - 3.65T + 11T^{2} \)
13 \( 1 + 0.859iT - 13T^{2} \)
17 \( 1 - 6.72iT - 17T^{2} \)
19 \( 1 - 1.51T + 19T^{2} \)
29 \( 1 + 0.548T + 29T^{2} \)
31 \( 1 + 5.99T + 31T^{2} \)
37 \( 1 - 2.04iT - 37T^{2} \)
41 \( 1 + 7.14T + 41T^{2} \)
43 \( 1 - 10.0iT - 43T^{2} \)
47 \( 1 + 9.17iT - 47T^{2} \)
53 \( 1 + 5.37iT - 53T^{2} \)
59 \( 1 - 0.582T + 59T^{2} \)
61 \( 1 + 8.83T + 61T^{2} \)
67 \( 1 + 3.20iT - 67T^{2} \)
71 \( 1 + 12.5T + 71T^{2} \)
73 \( 1 + 8.62iT - 73T^{2} \)
79 \( 1 + 0.0700T + 79T^{2} \)
83 \( 1 + 6.74iT - 83T^{2} \)
89 \( 1 + 4.96T + 89T^{2} \)
97 \( 1 - 11.3iT - 97T^{2} \)
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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

−13.22302752087141225259470214969, −12.20835236617289355040834468569, −11.63129517410701883040767561344, −10.68502782557959940962325166333, −8.465921402527068332540631328356, −7.61096907042886788978352739538, −6.86621543059402522395376515059, −6.06854828442604128966245026174, −3.41639503190700492952070637178, −1.60887409071725346005761753544, 3.12281663194692789890452688853, 4.23434331060069486745540495821, 5.51628664312913096544071143333, 7.19325903841520080124046455794, 8.809017901812394614155517229268, 9.527140712732078374397381465580, 10.78843309953419676647018045758, 11.59097793107749584221307247328, 12.11411480481556108869085261254, 14.18864712208623600448331190968

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