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Properties

Label 2-28e2-112.75-c1-0-55
Degree $2$
Conductor $784$
Sign $0.954 - 0.297i$
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  + (1.40 + 0.146i)2-s + (1.76 + 0.472i)3-s + (1.95 + 0.412i)4-s + (−0.219 − 0.818i)5-s + (2.40 + 0.922i)6-s + (2.69 + 0.867i)8-s + (0.284 + 0.164i)9-s + (−0.188 − 1.18i)10-s + (2.73 + 0.732i)11-s + (3.25 + 1.65i)12-s + (1.00 + 1.00i)13-s − 1.54i·15-s + (3.65 + 1.61i)16-s + (−5.61 + 3.24i)17-s + (0.375 + 0.272i)18-s + (−1.43 − 5.37i)19-s + ⋯
L(s)  = 1  + (0.994 + 0.103i)2-s + (1.01 + 0.272i)3-s + (0.978 + 0.206i)4-s + (−0.0981 − 0.366i)5-s + (0.983 + 0.376i)6-s + (0.951 + 0.306i)8-s + (0.0947 + 0.0546i)9-s + (−0.0595 − 0.374i)10-s + (0.823 + 0.220i)11-s + (0.939 + 0.476i)12-s + (0.277 + 0.277i)13-s − 0.399i·15-s + (0.914 + 0.403i)16-s + (−1.36 + 0.785i)17-s + (0.0885 + 0.0642i)18-s + (−0.330 − 1.23i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.954 - 0.297i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.954 - 0.297i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $0.954 - 0.297i$
Analytic conductor: \(6.26027\)
Root analytic conductor: \(2.50205\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (411, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :1/2),\ 0.954 - 0.297i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.84469 + 0.585241i\)
\(L(\frac12)\) \(\approx\) \(3.84469 + 0.585241i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.40 - 0.146i)T \)
7 \( 1 \)
good3 \( 1 + (-1.76 - 0.472i)T + (2.59 + 1.5i)T^{2} \)
5 \( 1 + (0.219 + 0.818i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (-2.73 - 0.732i)T + (9.52 + 5.5i)T^{2} \)
13 \( 1 + (-1.00 - 1.00i)T + 13iT^{2} \)
17 \( 1 + (5.61 - 3.24i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.43 + 5.37i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (2.20 - 3.81i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.241 + 0.241i)T - 29iT^{2} \)
31 \( 1 + (1.69 + 2.93i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.73 - 1.00i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 4.88T + 41T^{2} \)
43 \( 1 + (-4.40 + 4.40i)T - 43iT^{2} \)
47 \( 1 + (4.72 - 8.18i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.17 + 4.39i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-0.494 + 1.84i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (13.4 - 3.60i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (3.18 - 11.9i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 12.2T + 71T^{2} \)
73 \( 1 + (-0.402 - 0.697i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (9.32 + 5.38i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.76 + 5.76i)T - 83iT^{2} \)
89 \( 1 + (-6.66 + 11.5i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 10.8iT - 97T^{2} \)
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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

−10.46486141860566728652157885223, −9.142449164290607646485224943460, −8.790839100466699956570343470394, −7.72397551708586075150732906207, −6.73234545271568943230875277082, −5.94330607440929402024736909053, −4.52747447900962937231619155118, −4.05571851315923214266409677054, −2.93316995591996270404493978026, −1.85964729146667180833162738133, 1.75747965486032350986112137191, 2.80971077688196536763587658439, 3.63067571720983098961968698613, 4.61627369664982854397657669988, 5.90475071577382540757548226022, 6.72198036339743754296863873397, 7.54259716866418214109012346854, 8.495125068980612151161440922258, 9.271416186447559204541754651089, 10.53071540359631636927681626603

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