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Properties

Label 2-14e2-7.4-c3-0-5
Degree $2$
Conductor $196$
Sign $0.968 + 0.250i$
Analytic cond. $11.5643$
Root an. cond. $3.40064$
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  + (2 − 3.46i)3-s + (3 + 5.19i)5-s + (5.50 + 9.52i)9-s + (6 − 10.3i)11-s + 82·13-s + 24·15-s + (−15 + 25.9i)17-s + (34 + 58.8i)19-s + (−108 − 187. i)23-s + (44.5 − 77.0i)25-s + 152·27-s + 246·29-s + (−56 + 96.9i)31-s + (−24 − 41.5i)33-s + (−55 − 95.2i)37-s + ⋯
L(s)  = 1  + (0.384 − 0.666i)3-s + (0.268 + 0.464i)5-s + (0.203 + 0.352i)9-s + (0.164 − 0.284i)11-s + 1.74·13-s + 0.413·15-s + (−0.214 + 0.370i)17-s + (0.410 + 0.711i)19-s + (−0.979 − 1.69i)23-s + (0.355 − 0.616i)25-s + 1.08·27-s + 1.57·29-s + (−0.324 + 0.561i)31-s + (−0.126 − 0.219i)33-s + (−0.244 − 0.423i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.250i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.968 + 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(196\)    =    \(2^{2} \cdot 7^{2}\)
Sign: $0.968 + 0.250i$
Analytic conductor: \(11.5643\)
Root analytic conductor: \(3.40064\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{196} (165, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 196,\ (\ :3/2),\ 0.968 + 0.250i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.24452 - 0.286027i\)
\(L(\frac12)\) \(\approx\) \(2.24452 - 0.286027i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-2 + 3.46i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 + (-3 - 5.19i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-6 + 10.3i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 82T + 2.19e3T^{2} \)
17 \( 1 + (15 - 25.9i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-34 - 58.8i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (108 + 187. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 246T + 2.43e4T^{2} \)
31 \( 1 + (56 - 96.9i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (55 + 95.2i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 246T + 6.89e4T^{2} \)
43 \( 1 + 172T + 7.95e4T^{2} \)
47 \( 1 + (-96 - 166. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (279 - 483. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-270 + 467. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-55 - 95.2i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (70 - 121. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 840T + 3.57e5T^{2} \)
73 \( 1 + (275 - 476. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-104 - 180. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 516T + 5.71e5T^{2} \)
89 \( 1 + (699 + 1.21e3i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 1.58e3T + 9.12e5T^{2} \)
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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

−12.19104186076367065262052749083, −10.84535667531398660087712899558, −10.27746064040980111240611151560, −8.666965902643462388370922549174, −8.109848321054145705857198499253, −6.73122852067868339378403309839, −6.01379925428326214313490033173, −4.22860721638268050654126096833, −2.74916648706592522728312108179, −1.32563164097220995081119533531, 1.28683019263644762006400097334, 3.28917058948117662746066198757, 4.34523762567819144033007575638, 5.65250517526696986807970729549, 6.88116742067269217856129563980, 8.335112347241270824984279830211, 9.187675059109256076965301748889, 9.885628396754559146360375486759, 11.07572653628383418502379006628, 12.00998268459671747447274392599

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