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Properties

Label 2-28e2-196.111-c1-0-0
Degree $2$
Conductor $784$
Sign $-0.590 - 0.806i$
Analytic cond. $6.26027$
Root an. cond. $2.50205$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.977 − 1.22i)3-s + (−0.470 + 0.374i)5-s + (1.48 + 2.19i)7-s + (0.120 − 0.529i)9-s + (−5.53 + 1.26i)11-s + (0.517 − 0.118i)13-s + (0.918 + 0.209i)15-s + (0.00663 − 0.0137i)17-s − 4.54·19-s + (1.24 − 3.95i)21-s + (0.102 + 0.212i)23-s + (−1.03 + 4.52i)25-s + (−5.00 + 2.40i)27-s + (−3.07 − 1.48i)29-s − 5.84·31-s + ⋯
L(s)  = 1  + (−0.564 − 0.707i)3-s + (−0.210 + 0.167i)5-s + (0.559 + 0.828i)7-s + (0.0402 − 0.176i)9-s + (−1.66 + 0.380i)11-s + (0.143 − 0.0327i)13-s + (0.237 + 0.0541i)15-s + (0.00160 − 0.00334i)17-s − 1.04·19-s + (0.270 − 0.863i)21-s + (0.0213 + 0.0443i)23-s + (−0.206 + 0.904i)25-s + (−0.962 + 0.463i)27-s + (−0.571 − 0.275i)29-s − 1.05·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.590 - 0.806i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.590 - 0.806i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $-0.590 - 0.806i$
Analytic conductor: \(6.26027\)
Root analytic conductor: \(2.50205\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (111, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :1/2),\ -0.590 - 0.806i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.153069 + 0.301786i\)
\(L(\frac12)\) \(\approx\) \(0.153069 + 0.301786i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.48 - 2.19i)T \)
good3 \( 1 + (0.977 + 1.22i)T + (-0.667 + 2.92i)T^{2} \)
5 \( 1 + (0.470 - 0.374i)T + (1.11 - 4.87i)T^{2} \)
11 \( 1 + (5.53 - 1.26i)T + (9.91 - 4.77i)T^{2} \)
13 \( 1 + (-0.517 + 0.118i)T + (11.7 - 5.64i)T^{2} \)
17 \( 1 + (-0.00663 + 0.0137i)T + (-10.5 - 13.2i)T^{2} \)
19 \( 1 + 4.54T + 19T^{2} \)
23 \( 1 + (-0.102 - 0.212i)T + (-14.3 + 17.9i)T^{2} \)
29 \( 1 + (3.07 + 1.48i)T + (18.0 + 22.6i)T^{2} \)
31 \( 1 + 5.84T + 31T^{2} \)
37 \( 1 + (-5.35 - 2.57i)T + (23.0 + 28.9i)T^{2} \)
41 \( 1 + (-1.23 + 0.983i)T + (9.12 - 39.9i)T^{2} \)
43 \( 1 + (-5.47 - 4.36i)T + (9.56 + 41.9i)T^{2} \)
47 \( 1 + (0.699 + 3.06i)T + (-42.3 + 20.3i)T^{2} \)
53 \( 1 + (10.7 - 5.16i)T + (33.0 - 41.4i)T^{2} \)
59 \( 1 + (6.17 - 7.74i)T + (-13.1 - 57.5i)T^{2} \)
61 \( 1 + (3.48 - 7.23i)T + (-38.0 - 47.6i)T^{2} \)
67 \( 1 - 5.76iT - 67T^{2} \)
71 \( 1 + (-1.23 - 2.56i)T + (-44.2 + 55.5i)T^{2} \)
73 \( 1 + (-3.26 - 0.746i)T + (65.7 + 31.6i)T^{2} \)
79 \( 1 - 7.84iT - 79T^{2} \)
83 \( 1 + (-3.22 + 14.1i)T + (-74.7 - 36.0i)T^{2} \)
89 \( 1 + (4.67 + 1.06i)T + (80.1 + 38.6i)T^{2} \)
97 \( 1 + 13.2iT - 97T^{2} \)
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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

−10.92511262928697861147565353308, −9.754325507376184471852039807265, −8.804054215309814648037011509222, −7.80353168834913760030590572596, −7.29654980611435745523876566158, −6.05350690026534262154543735541, −5.50481654273026906516467890346, −4.39263130541990207236901426999, −2.85823845398561624457004074458, −1.73952713812853878780610850781, 0.17229743406547561571602744538, 2.18024787532486205952577481623, 3.73712839065846033040431959096, 4.66432342388976133436449766395, 5.28502118590579571444287446519, 6.34200885806404043847467802123, 7.77681768339541169083089313152, 7.970821671113377419073901566617, 9.286088236660948592200255504425, 10.30664555180966978747337119015

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