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Properties

Label 2-51-51.38-c4-0-17
Degree $2$
Conductor $51$
Sign $0.994 + 0.108i$
Analytic cond. $5.27186$
Root an. cond. $2.29605$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 6.32·2-s + (8.98 + 0.488i)3-s + 24.0·4-s + (−17.5 − 17.5i)5-s + (56.8 + 3.08i)6-s + (2.97 + 2.97i)7-s + 50.6·8-s + (80.5 + 8.77i)9-s + (−111. − 111. i)10-s + (−162. + 162. i)11-s + (215. + 11.7i)12-s + 3.77·13-s + (18.8 + 18.8i)14-s + (−149. − 166. i)15-s − 63.9·16-s + (217. + 189. i)17-s + ⋯
L(s)  = 1  + 1.58·2-s + (0.998 + 0.0542i)3-s + 1.50·4-s + (−0.702 − 0.702i)5-s + (1.57 + 0.0857i)6-s + (0.0608 + 0.0608i)7-s + 0.790·8-s + (0.994 + 0.108i)9-s + (−1.11 − 1.11i)10-s + (−1.34 + 1.34i)11-s + (1.49 + 0.0813i)12-s + 0.0223·13-s + (0.0961 + 0.0961i)14-s + (−0.662 − 0.739i)15-s − 0.249·16-s + (0.753 + 0.657i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.108i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.994 + 0.108i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(51\)    =    \(3 \cdot 17\)
Sign: $0.994 + 0.108i$
Analytic conductor: \(5.27186\)
Root analytic conductor: \(2.29605\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{51} (38, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 51,\ (\ :2),\ 0.994 + 0.108i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(3.92535 - 0.212883i\)
\(L(\frac12)\) \(\approx\) \(3.92535 - 0.212883i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-8.98 - 0.488i)T \)
17 \( 1 + (-217. - 189. i)T \)
good2 \( 1 - 6.32T + 16T^{2} \)
5 \( 1 + (17.5 + 17.5i)T + 625iT^{2} \)
7 \( 1 + (-2.97 - 2.97i)T + 2.40e3iT^{2} \)
11 \( 1 + (162. - 162. i)T - 1.46e4iT^{2} \)
13 \( 1 - 3.77T + 2.85e4T^{2} \)
19 \( 1 + 434. iT - 1.30e5T^{2} \)
23 \( 1 + (-40.0 + 40.0i)T - 2.79e5iT^{2} \)
29 \( 1 + (188. + 188. i)T + 7.07e5iT^{2} \)
31 \( 1 + (-562. + 562. i)T - 9.23e5iT^{2} \)
37 \( 1 + (-243. + 243. i)T - 1.87e6iT^{2} \)
41 \( 1 + (1.77e3 - 1.77e3i)T - 2.82e6iT^{2} \)
43 \( 1 - 316. iT - 3.41e6T^{2} \)
47 \( 1 + 318. iT - 4.87e6T^{2} \)
53 \( 1 - 3.97e3T + 7.89e6T^{2} \)
59 \( 1 - 3.78e3T + 1.21e7T^{2} \)
61 \( 1 + (-4.98e3 - 4.98e3i)T + 1.38e7iT^{2} \)
67 \( 1 - 3.57e3T + 2.01e7T^{2} \)
71 \( 1 + (5.33e3 + 5.33e3i)T + 2.54e7iT^{2} \)
73 \( 1 + (3.11e3 - 3.11e3i)T - 2.83e7iT^{2} \)
79 \( 1 + (3.47e3 + 3.47e3i)T + 3.89e7iT^{2} \)
83 \( 1 + 4.52e3T + 4.74e7T^{2} \)
89 \( 1 - 9.56e3iT - 6.27e7T^{2} \)
97 \( 1 + (9.23e3 - 9.23e3i)T - 8.85e7iT^{2} \)
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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

−14.87313849774168328628588594055, −13.35237318201482486874358573460, −12.87082477334287040598336425272, −11.83409561525090799110779781043, −10.06545810906070450088389774795, −8.375671990483352443913052822139, −7.18122973370403928828023842592, −5.07815139091317041336858794425, −4.11966571502622728201469744370, −2.52412373974767328990860630440, 2.87094264489679019756133395956, 3.69227111371765164389690934634, 5.46846208764587710646323543008, 7.16318201651814106495498754497, 8.308125604750589660571435966680, 10.36756730718097648459044747905, 11.61662838450638616780800641260, 12.83603261372069112577705642029, 13.83028499435826390672344865248, 14.46587501791239694665623711878

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