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Properties

Label 2-28e2-112.109-c1-0-2
Degree $2$
Conductor $784$
Sign $-0.126 - 0.991i$
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  + (0.545 − 1.30i)2-s + (0.569 − 0.152i)3-s + (−1.40 − 1.42i)4-s + (−1.54 − 0.414i)5-s + (0.111 − 0.826i)6-s + (−2.62 + 1.05i)8-s + (−2.29 + 1.32i)9-s + (−1.38 + 1.79i)10-s + (1.41 + 5.26i)11-s + (−1.01 − 0.595i)12-s + (−4.66 − 4.66i)13-s − 0.943·15-s + (−0.0498 + 3.99i)16-s + (−2.66 + 4.61i)17-s + (0.478 + 3.72i)18-s + (0.936 − 3.49i)19-s + ⋯
L(s)  = 1  + (0.385 − 0.922i)2-s + (0.328 − 0.0880i)3-s + (−0.702 − 0.711i)4-s + (−0.691 − 0.185i)5-s + (0.0454 − 0.337i)6-s + (−0.927 + 0.374i)8-s + (−0.765 + 0.442i)9-s + (−0.437 + 0.566i)10-s + (0.425 + 1.58i)11-s + (−0.293 − 0.172i)12-s + (−1.29 − 1.29i)13-s − 0.243·15-s + (−0.0124 + 0.999i)16-s + (−0.646 + 1.12i)17-s + (0.112 + 0.876i)18-s + (0.214 − 0.802i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.126 - 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.126 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $-0.126 - 0.991i$
Analytic conductor: \(6.26027\)
Root analytic conductor: \(2.50205\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (557, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :1/2),\ -0.126 - 0.991i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0700065 + 0.0794926i\)
\(L(\frac12)\) \(\approx\) \(0.0700065 + 0.0794926i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.545 + 1.30i)T \)
7 \( 1 \)
good3 \( 1 + (-0.569 + 0.152i)T + (2.59 - 1.5i)T^{2} \)
5 \( 1 + (1.54 + 0.414i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (-1.41 - 5.26i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (4.66 + 4.66i)T + 13iT^{2} \)
17 \( 1 + (2.66 - 4.61i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.936 + 3.49i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (2.25 - 1.30i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.22 + 1.22i)T + 29iT^{2} \)
31 \( 1 + (0.416 - 0.722i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-6.05 - 1.62i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 0.263iT - 41T^{2} \)
43 \( 1 + (-1.25 + 1.25i)T - 43iT^{2} \)
47 \( 1 + (5.37 + 9.31i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.0174 - 0.0650i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-1.32 - 4.92i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (1.63 - 6.09i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (12.9 - 3.48i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 2.05iT - 71T^{2} \)
73 \( 1 + (4.74 + 2.74i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.60 + 4.51i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.84 + 5.84i)T + 83iT^{2} \)
89 \( 1 + (-5.47 + 3.16i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 18.8T + 97T^{2} \)
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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

−10.46641718059613745328935947950, −9.894349999703933636272059756316, −8.939700657641020943485739681671, −8.053819910381409348417800188659, −7.26368044586830699164980573476, −5.84497185838194205754996468114, −4.83854673507546765969985828237, −4.12653243413161928708718735348, −2.87687525753133603071321196369, −1.97057836801979158980215128494, 0.04218484790488451035407851106, 2.78525756817606586549889258701, 3.70338964832157661371601106169, 4.58110426647118173907604423136, 5.78470491035675993372558371921, 6.53945668890795208790119121818, 7.50075763074941330425000844611, 8.210565074202256091524429535498, 9.121667893512948645758644937807, 9.566080296300352952684658703512

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