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Properties

Label 2-114-19.7-c3-0-2
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: $\chi_{114} (7, \cdot )$
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

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

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