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Properties

Label 2-114-19.11-c3-0-5
Degree $2$
Conductor $114$
Sign $0.321 + 0.946i$
Analytic cond. $6.72621$
Root an. cond. $2.59349$
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  + (1 − 1.73i)2-s + (−1.5 + 2.59i)3-s + (−1.99 − 3.46i)4-s + (3 − 5.19i)5-s + (3 + 5.19i)6-s + 19·7-s − 7.99·8-s + (−4.5 − 7.79i)9-s + (−6 − 10.3i)10-s + 32·11-s + 12·12-s + (−40.5 − 70.1i)13-s + (19 − 32.9i)14-s + (9 + 15.5i)15-s + (−8 + 13.8i)16-s + (62 − 107. i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.288 + 0.499i)3-s + (−0.249 − 0.433i)4-s + (0.268 − 0.464i)5-s + (0.204 + 0.353i)6-s + 1.02·7-s − 0.353·8-s + (−0.166 − 0.288i)9-s + (−0.189 − 0.328i)10-s + 0.877·11-s + 0.288·12-s + (−0.864 − 1.49i)13-s + (0.362 − 0.628i)14-s + (0.154 + 0.268i)15-s + (−0.125 + 0.216i)16-s + (0.884 − 1.53i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.321 + 0.946i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.321 + 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(114\)    =    \(2 \cdot 3 \cdot 19\)
Sign: $0.321 + 0.946i$
Analytic conductor: \(6.72621\)
Root analytic conductor: \(2.59349\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{114} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 114,\ (\ :3/2),\ 0.321 + 0.946i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.55319 - 1.11233i\)
\(L(\frac12)\) \(\approx\) \(1.55319 - 1.11233i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 + 1.73i)T \)
3 \( 1 + (1.5 - 2.59i)T \)
19 \( 1 + (-76 - 32.9i)T \)
good5 \( 1 + (-3 + 5.19i)T + (-62.5 - 108. i)T^{2} \)
7 \( 1 - 19T + 343T^{2} \)
11 \( 1 - 32T + 1.33e3T^{2} \)
13 \( 1 + (40.5 + 70.1i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (-62 + 107. i)T + (-2.45e3 - 4.25e3i)T^{2} \)
23 \( 1 + (49 + 84.8i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-150 - 259. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + 225T + 2.97e4T^{2} \)
37 \( 1 + 293T + 5.06e4T^{2} \)
41 \( 1 + (88 - 152. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-55.5 + 96.1i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-275 - 476. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-241 - 417. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-248 + 429. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (77.5 + 134. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (232.5 + 402. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (-55 + 95.2i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + (408.5 - 707. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (129.5 - 224. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 56T + 5.71e5T^{2} \)
89 \( 1 + (154 + 266. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-575 + 995. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−12.56258293892930679305856181152, −11.96398291245774499534439382832, −10.86108852955366862138181329533, −9.894846658558873318102411398968, −8.882350075470621731292993407723, −7.42212554186223753219779274288, −5.42978983267366805312803622982, −4.89785958207589361124280235533, −3.17677895364682413284194971937, −1.11349394767555757514763443247, 1.84185130399854925992129792155, 4.06193148018433856107860322379, 5.46874937762295184959134129730, 6.64797123327613201598483087889, 7.55796333114033075232817546366, 8.773288457154388514962806411513, 10.16224536488760939362466459725, 11.67470675064708984843769124231, 12.07482926821236752785667140353, 13.65367864567486072247173983925

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