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Properties

Label 2-115-115.114-c2-0-9
Degree $2$
Conductor $115$
Sign $0.794 + 0.606i$
Analytic cond. $3.13352$
Root an. cond. $1.77017$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.10i·2-s + 1.12i·3-s − 0.431·4-s + (0.993 + 4.90i)5-s + 2.37·6-s + 4.72·7-s − 7.51i·8-s + 7.72·9-s + (10.3 − 2.09i)10-s + 8.84i·11-s − 0.486i·12-s − 8.75i·13-s − 9.94i·14-s + (−5.52 + 1.11i)15-s − 17.5·16-s + 1.51·17-s + ⋯
L(s)  = 1  − 1.05i·2-s + 0.375i·3-s − 0.107·4-s + (0.198 + 0.980i)5-s + 0.395·6-s + 0.674·7-s − 0.939i·8-s + 0.858·9-s + (1.03 − 0.209i)10-s + 0.803i·11-s − 0.0405i·12-s − 0.673i·13-s − 0.710i·14-s + (−0.368 + 0.0746i)15-s − 1.09·16-s + 0.0893·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.794 + 0.606i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.794 + 0.606i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $0.794 + 0.606i$
Analytic conductor: \(3.13352\)
Root analytic conductor: \(1.77017\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{115} (114, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 115,\ (\ :1),\ 0.794 + 0.606i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.57457 - 0.532455i\)
\(L(\frac12)\) \(\approx\) \(1.57457 - 0.532455i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.993 - 4.90i)T \)
23 \( 1 + (17.3 - 15.1i)T \)
good2 \( 1 + 2.10iT - 4T^{2} \)
3 \( 1 - 1.12iT - 9T^{2} \)
7 \( 1 - 4.72T + 49T^{2} \)
11 \( 1 - 8.84iT - 121T^{2} \)
13 \( 1 + 8.75iT - 169T^{2} \)
17 \( 1 - 1.51T + 289T^{2} \)
19 \( 1 + 19.7iT - 361T^{2} \)
29 \( 1 - 27.8T + 841T^{2} \)
31 \( 1 - 4.05T + 961T^{2} \)
37 \( 1 + 49.6T + 1.36e3T^{2} \)
41 \( 1 - 15.8T + 1.68e3T^{2} \)
43 \( 1 + 35.4T + 1.84e3T^{2} \)
47 \( 1 - 33.6iT - 2.20e3T^{2} \)
53 \( 1 + 29.2T + 2.80e3T^{2} \)
59 \( 1 + 57.1T + 3.48e3T^{2} \)
61 \( 1 + 96.2iT - 3.72e3T^{2} \)
67 \( 1 - 89.7T + 4.48e3T^{2} \)
71 \( 1 + 36.1T + 5.04e3T^{2} \)
73 \( 1 + 117. iT - 5.32e3T^{2} \)
79 \( 1 + 46.4iT - 6.24e3T^{2} \)
83 \( 1 + 114.T + 6.88e3T^{2} \)
89 \( 1 - 65.0iT - 7.92e3T^{2} \)
97 \( 1 - 138.T + 9.40e3T^{2} \)
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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

−13.00034063809204154575515478749, −11.98271244810663151001474973998, −10.98763292192500154623206649297, −10.28521095787825518501829108385, −9.538998736731468594653083733502, −7.65245373654233891884170219093, −6.62865445210276022912102936444, −4.72464280651497266818809304013, −3.30798285602384196707312629458, −1.85671782670320943634283371837, 1.68838603386496111596497709009, 4.44340784518747994706602833587, 5.66095650897281541019411812305, 6.75778018373177650250212366232, 8.018260645377108174311992231872, 8.615401163523863524756621072121, 10.14889010125251672885136723406, 11.59720285379082121605806449767, 12.45574367752788858893771096912, 13.74684247804967598601892887246

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