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Properties

Label 2-115-5.3-c2-0-6
Degree $2$
Conductor $115$
Sign $-0.977 - 0.209i$
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  + (−2.74 + 2.74i)2-s + (3.72 + 3.72i)3-s − 11.0i·4-s + (4.51 + 2.14i)5-s − 20.4·6-s + (−2.87 + 2.87i)7-s + (19.4 + 19.4i)8-s + 18.7i·9-s + (−18.2 + 6.52i)10-s − 0.852·11-s + (41.3 − 41.3i)12-s + (−2.46 − 2.46i)13-s − 15.8i·14-s + (8.85 + 24.8i)15-s − 62.6·16-s + (4.78 − 4.78i)17-s + ⋯
L(s)  = 1  + (−1.37 + 1.37i)2-s + (1.24 + 1.24i)3-s − 2.77i·4-s + (0.903 + 0.428i)5-s − 3.41·6-s + (−0.410 + 0.410i)7-s + (2.43 + 2.43i)8-s + 2.08i·9-s + (−1.82 + 0.652i)10-s − 0.0775·11-s + (3.44 − 3.44i)12-s + (−0.189 − 0.189i)13-s − 1.12i·14-s + (0.590 + 1.65i)15-s − 3.91·16-s + (0.281 − 0.281i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.977 - 0.209i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.977 - 0.209i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $-0.977 - 0.209i$
Analytic conductor: \(3.13352\)
Root analytic conductor: \(1.77017\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{115} (93, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 115,\ (\ :1),\ -0.977 - 0.209i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.115970 + 1.09599i\)
\(L(\frac12)\) \(\approx\) \(0.115970 + 1.09599i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-4.51 - 2.14i)T \)
23 \( 1 + (3.39 + 3.39i)T \)
good2 \( 1 + (2.74 - 2.74i)T - 4iT^{2} \)
3 \( 1 + (-3.72 - 3.72i)T + 9iT^{2} \)
7 \( 1 + (2.87 - 2.87i)T - 49iT^{2} \)
11 \( 1 + 0.852T + 121T^{2} \)
13 \( 1 + (2.46 + 2.46i)T + 169iT^{2} \)
17 \( 1 + (-4.78 + 4.78i)T - 289iT^{2} \)
19 \( 1 + 17.8iT - 361T^{2} \)
29 \( 1 + 32.6iT - 841T^{2} \)
31 \( 1 - 18.9T + 961T^{2} \)
37 \( 1 + (12.7 - 12.7i)T - 1.36e3iT^{2} \)
41 \( 1 - 35.8T + 1.68e3T^{2} \)
43 \( 1 + (22.9 + 22.9i)T + 1.84e3iT^{2} \)
47 \( 1 + (12.3 - 12.3i)T - 2.20e3iT^{2} \)
53 \( 1 + (-54.5 - 54.5i)T + 2.80e3iT^{2} \)
59 \( 1 + 29.2iT - 3.48e3T^{2} \)
61 \( 1 + 89.6T + 3.72e3T^{2} \)
67 \( 1 + (-49.4 + 49.4i)T - 4.48e3iT^{2} \)
71 \( 1 - 50.3T + 5.04e3T^{2} \)
73 \( 1 + (-44.2 - 44.2i)T + 5.32e3iT^{2} \)
79 \( 1 + 109. iT - 6.24e3T^{2} \)
83 \( 1 + (2.16 + 2.16i)T + 6.88e3iT^{2} \)
89 \( 1 - 127. iT - 7.92e3T^{2} \)
97 \( 1 + (-131. + 131. i)T - 9.40e3iT^{2} \)
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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

−14.22645337888585872344111070354, −13.62199041867082090417534549028, −10.84033884880063333899740373564, −9.958483852829000264812407021622, −9.436948108083321340632839349620, −8.677693669190336234992633762605, −7.56185082074291446325948104171, −6.19427271382875297219580440230, −4.94647002325435909015416157079, −2.52351684183889704783446769857, 1.20285157838158390777288015989, 2.26498282951109931363759480206, 3.52967904947834466225356395679, 6.77299970968908215572115246667, 7.87237534662338508764589348553, 8.697282259777986738264100409465, 9.549459589494558208605906587619, 10.37876730748015369959790034311, 12.00576376320682698525503822906, 12.75557640443402756846021778879

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