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Properties

Label 2-14e2-7.2-c3-0-3
Degree $2$
Conductor $196$
Sign $-0.386 - 0.922i$
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  + (5 + 8.66i)3-s + (4 − 6.92i)5-s + (−36.5 + 63.2i)9-s + (20 + 34.6i)11-s − 12·13-s + 80·15-s + (29 + 50.2i)17-s + (−13 + 22.5i)19-s + (32 − 55.4i)23-s + (30.5 + 52.8i)25-s − 460.·27-s − 62·29-s + (−126 − 218. i)31-s + (−200. + 346. i)33-s + (−13 + 22.5i)37-s + ⋯
L(s)  = 1  + (0.962 + 1.66i)3-s + (0.357 − 0.619i)5-s + (−1.35 + 2.34i)9-s + (0.548 + 0.949i)11-s − 0.256·13-s + 1.37·15-s + (0.413 + 0.716i)17-s + (−0.156 + 0.271i)19-s + (0.290 − 0.502i)23-s + (0.244 + 0.422i)25-s − 3.27·27-s − 0.397·29-s + (−0.730 − 1.26i)31-s + (−1.05 + 1.82i)33-s + (−0.0577 + 0.100i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(196\)    =    \(2^{2} \cdot 7^{2}\)
Sign: $-0.386 - 0.922i$
Analytic conductor: \(11.5643\)
Root analytic conductor: \(3.40064\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{196} (177, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 196,\ (\ :3/2),\ -0.386 - 0.922i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.32013 + 1.98461i\)
\(L(\frac12)\) \(\approx\) \(1.32013 + 1.98461i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-5 - 8.66i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 + (-4 + 6.92i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-20 - 34.6i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 12T + 2.19e3T^{2} \)
17 \( 1 + (-29 - 50.2i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (13 - 22.5i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-32 + 55.4i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 62T + 2.43e4T^{2} \)
31 \( 1 + (126 + 218. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (13 - 22.5i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 6T + 6.89e4T^{2} \)
43 \( 1 - 416T + 7.95e4T^{2} \)
47 \( 1 + (-198 + 342. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-225 - 389. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (137 + 237. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-288 + 498. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-238 - 412. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 448T + 3.57e5T^{2} \)
73 \( 1 + (-79 - 136. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-468 + 810. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 530T + 5.71e5T^{2} \)
89 \( 1 + (-195 + 337. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 214T + 9.12e5T^{2} \)
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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

−12.43755530188440614928341334468, −11.03710599351132318421050379612, −10.10633275032607655611204346792, −9.385430617173997507533918460550, −8.737257793297386489937285922803, −7.57970861253101630942520882596, −5.64568051608364021810242730611, −4.58451412592688943143243306851, −3.72429166374827658842227593687, −2.15501281059897491347571402892, 0.976235712202319232569156427841, 2.43235131797311039427752703867, 3.39983951253510365262967404795, 5.79841274672679197589537583175, 6.79487698336805675792199409554, 7.49417516315306627491714745874, 8.625963226495651196041468102828, 9.390649417109911477062086456869, 10.96315528129379905217966183099, 11.98470222142444301110988160710

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