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Properties

Label 2-115-23.22-c2-0-1
Degree $2$
Conductor $115$
Sign $-0.597 - 0.802i$
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-s + 0.396·3-s − 3·4-s − 2.23i·5-s − 0.396·6-s + 8.16i·7-s + 7·8-s − 8.84·9-s + 2.23i·10-s + 18.9i·11-s − 1.18·12-s − 18.6·13-s − 8.16i·14-s − 0.885i·15-s + 5·16-s − 5.57i·17-s + ⋯
L(s)  = 1  − 0.5·2-s + 0.132·3-s − 0.750·4-s − 0.447i·5-s − 0.0660·6-s + 1.16i·7-s + 0.875·8-s − 0.982·9-s + 0.223i·10-s + 1.72i·11-s − 0.0990·12-s − 1.43·13-s − 0.583i·14-s − 0.0590i·15-s + 0.312·16-s − 0.327i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.597 - 0.802i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.597 - 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $-0.597 - 0.802i$
Analytic conductor: \(3.13352\)
Root analytic conductor: \(1.77017\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{115} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 115,\ (\ :1),\ -0.597 - 0.802i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.212670 + 0.423538i\)
\(L(\frac12)\) \(\approx\) \(0.212670 + 0.423538i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + 2.23iT \)
23 \( 1 + (18.4 - 13.7i)T \)
good2 \( 1 + T + 4T^{2} \)
3 \( 1 - 0.396T + 9T^{2} \)
7 \( 1 - 8.16iT - 49T^{2} \)
11 \( 1 - 18.9iT - 121T^{2} \)
13 \( 1 + 18.6T + 169T^{2} \)
17 \( 1 + 5.57iT - 289T^{2} \)
19 \( 1 - 5.25iT - 361T^{2} \)
29 \( 1 - 11.6T + 841T^{2} \)
31 \( 1 - 49.5T + 961T^{2} \)
37 \( 1 + 29.7iT - 1.36e3T^{2} \)
41 \( 1 + 10.4T + 1.68e3T^{2} \)
43 \( 1 + 57.2iT - 1.84e3T^{2} \)
47 \( 1 + 15.2T + 2.20e3T^{2} \)
53 \( 1 - 40.2iT - 2.80e3T^{2} \)
59 \( 1 + 27.7T + 3.48e3T^{2} \)
61 \( 1 - 110. iT - 3.72e3T^{2} \)
67 \( 1 + 35.0iT - 4.48e3T^{2} \)
71 \( 1 + 70.4T + 5.04e3T^{2} \)
73 \( 1 + 0.377T + 5.32e3T^{2} \)
79 \( 1 - 102. iT - 6.24e3T^{2} \)
83 \( 1 - 116. iT - 6.88e3T^{2} \)
89 \( 1 - 49.6iT - 7.92e3T^{2} \)
97 \( 1 + 69.6iT - 9.40e3T^{2} \)
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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

−13.75677672339866887875099155256, −12.34164426235018446153084098861, −12.01522109523490172893201210524, −10.07821953761592870143475109749, −9.413569887747486527880429154900, −8.498562252877243683532421923809, −7.44091616050910172740891091584, −5.52672603610774314416552750227, −4.56466438561201244349145874715, −2.31405819151971534130614912126, 0.38970212509846125275750858787, 3.16588245076992237593994637047, 4.68126401215102877797306390303, 6.28288298550766513242570000846, 7.80355792889313172818936540948, 8.521451356642158963922695634164, 9.856784683828807411552348718661, 10.63902523332006551031143120020, 11.71939088047803896402452945730, 13.30237057111521590518999548538

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