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Properties

Label 2-115-23.22-c6-0-28
Degree $2$
Conductor $115$
Sign $0.404 - 0.914i$
Analytic cond. $26.4562$
Root an. cond. $5.14356$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.74·2-s + 45.9·3-s − 31.0·4-s + 55.9i·5-s + 263.·6-s + 253. i·7-s − 545.·8-s + 1.37e3·9-s + 321. i·10-s + 1.23e3i·11-s − 1.42e3·12-s + 1.84e3·13-s + 1.45e3i·14-s + 2.56e3i·15-s − 1.14e3·16-s − 952. i·17-s + ⋯
L(s)  = 1  + 0.717·2-s + 1.70·3-s − 0.484·4-s + 0.447i·5-s + 1.22·6-s + 0.738i·7-s − 1.06·8-s + 1.89·9-s + 0.321i·10-s + 0.925i·11-s − 0.823·12-s + 0.840·13-s + 0.530i·14-s + 0.760i·15-s − 0.280·16-s − 0.193i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.404 - 0.914i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.404 - 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $0.404 - 0.914i$
Analytic conductor: \(26.4562\)
Root analytic conductor: \(5.14356\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{115} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 115,\ (\ :3),\ 0.404 - 0.914i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(4.182292936\)
\(L(\frac12)\) \(\approx\) \(4.182292936\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - 55.9iT \)
23 \( 1 + (-1.11e4 - 4.91e3i)T \)
good2 \( 1 - 5.74T + 64T^{2} \)
3 \( 1 - 45.9T + 729T^{2} \)
7 \( 1 - 253. iT - 1.17e5T^{2} \)
11 \( 1 - 1.23e3iT - 1.77e6T^{2} \)
13 \( 1 - 1.84e3T + 4.82e6T^{2} \)
17 \( 1 + 952. iT - 2.41e7T^{2} \)
19 \( 1 - 1.06e4iT - 4.70e7T^{2} \)
29 \( 1 + 403.T + 5.94e8T^{2} \)
31 \( 1 + 6.64e3T + 8.87e8T^{2} \)
37 \( 1 + 8.39e4iT - 2.56e9T^{2} \)
41 \( 1 + 6.40e4T + 4.75e9T^{2} \)
43 \( 1 + 1.00e5iT - 6.32e9T^{2} \)
47 \( 1 + 7.30e4T + 1.07e10T^{2} \)
53 \( 1 - 3.45e4iT - 2.21e10T^{2} \)
59 \( 1 - 3.56e5T + 4.21e10T^{2} \)
61 \( 1 - 3.15e5iT - 5.15e10T^{2} \)
67 \( 1 - 2.23e4iT - 9.04e10T^{2} \)
71 \( 1 - 1.40e5T + 1.28e11T^{2} \)
73 \( 1 + 4.01e5T + 1.51e11T^{2} \)
79 \( 1 + 7.32e5iT - 2.43e11T^{2} \)
83 \( 1 + 9.51e5iT - 3.26e11T^{2} \)
89 \( 1 - 3.91e4iT - 4.96e11T^{2} \)
97 \( 1 - 4.65e5iT - 8.32e11T^{2} \)
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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

−12.95398617814418943838709434815, −12.00321200326514080627903357552, −10.18415546942682403742204135528, −9.174504639748844909372031597405, −8.483897057257238669257519391154, −7.25061162395330008545567345661, −5.63427513727274398292263378872, −4.05904776397685994889068990811, −3.21117179021900056617536578722, −1.95256655776738242381399928827, 0.950125497159774301146735703643, 2.92834060967499147240415512370, 3.78636467155634232316238537559, 4.89184176282251596192379431267, 6.68904667326654428187812621493, 8.262610469272596866298478896086, 8.763243513312444269612463222183, 9.756081292654874457771125196325, 11.25588875309289688160499005203, 12.97934834492336589755442955759

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