Properties

Label 2-115-5.4-c3-0-30
Degree $2$
Conductor $115$
Sign $-0.900 - 0.434i$
Analytic cond. $6.78521$
Root an. cond. $2.60484$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4.24i·2-s − 2.36i·3-s − 10.0·4-s + (4.85 − 10.0i)5-s − 10.0·6-s − 12.8i·7-s + 8.69i·8-s + 21.3·9-s + (−42.7 − 20.6i)10-s − 45.1·11-s + 23.7i·12-s + 33.1i·13-s − 54.4·14-s + (−23.8 − 11.5i)15-s − 43.4·16-s + 61.8i·17-s + ⋯
L(s)  = 1  − 1.50i·2-s − 0.455i·3-s − 1.25·4-s + (0.434 − 0.900i)5-s − 0.684·6-s − 0.691i·7-s + 0.384i·8-s + 0.792·9-s + (−1.35 − 0.652i)10-s − 1.23·11-s + 0.572i·12-s + 0.708i·13-s − 1.03·14-s + (−0.410 − 0.197i)15-s − 0.678·16-s + 0.883i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.900 - 0.434i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.900 - 0.434i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $-0.900 - 0.434i$
Analytic conductor: \(6.78521\)
Root analytic conductor: \(2.60484\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{115} (24, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 115,\ (\ :3/2),\ -0.900 - 0.434i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.341475 + 1.49399i\)
\(L(\frac12)\) \(\approx\) \(0.341475 + 1.49399i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-4.85 + 10.0i)T \)
23 \( 1 + 23iT \)
good2 \( 1 + 4.24iT - 8T^{2} \)
3 \( 1 + 2.36iT - 27T^{2} \)
7 \( 1 + 12.8iT - 343T^{2} \)
11 \( 1 + 45.1T + 1.33e3T^{2} \)
13 \( 1 - 33.1iT - 2.19e3T^{2} \)
17 \( 1 - 61.8iT - 4.91e3T^{2} \)
19 \( 1 - 144.T + 6.85e3T^{2} \)
29 \( 1 + 145.T + 2.43e4T^{2} \)
31 \( 1 - 248.T + 2.97e4T^{2} \)
37 \( 1 - 71.9iT - 5.06e4T^{2} \)
41 \( 1 + 330.T + 6.89e4T^{2} \)
43 \( 1 + 356. iT - 7.95e4T^{2} \)
47 \( 1 + 338. iT - 1.03e5T^{2} \)
53 \( 1 + 464. iT - 1.48e5T^{2} \)
59 \( 1 - 645.T + 2.05e5T^{2} \)
61 \( 1 - 303.T + 2.26e5T^{2} \)
67 \( 1 - 365. iT - 3.00e5T^{2} \)
71 \( 1 - 6.02T + 3.57e5T^{2} \)
73 \( 1 + 319. iT - 3.89e5T^{2} \)
79 \( 1 + 2.86T + 4.93e5T^{2} \)
83 \( 1 - 1.34e3iT - 5.71e5T^{2} \)
89 \( 1 + 607.T + 7.04e5T^{2} \)
97 \( 1 - 1.09e3iT - 9.12e5T^{2} \)
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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

−12.45678715887292937335378358873, −11.62604854435190461214567529780, −10.25408891893752429689281019033, −9.825898219858691470306854009531, −8.387200122280160508095142955903, −7.02959976551918730394785116221, −5.13242293280321897653060751756, −3.86202343713446564157768788810, −2.04509965015638303352156538850, −0.870235680828497274854077027714, 2.87884594851822990627459212692, 5.02130111863948756673001856143, 5.78279518478099107532702513505, 7.14948880051362809446365269482, 7.85663150959806407537461763735, 9.374599010039191852913680531635, 10.19870868751563111802643775993, 11.50403683011590401487201851522, 13.13366593304561893277914410121, 13.94989983237218334998696207674

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