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Properties

Label 2-28e2-112.109-c1-0-5
Degree $2$
Conductor $784$
Sign $-0.383 - 0.923i$
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  + (−1.09 − 0.895i)2-s + (0.395 + 1.96i)4-s + (1.32 − 2.49i)8-s + (−2.59 + 1.5i)9-s + (0.236 + 0.882i)11-s + (−3.68 + 1.55i)16-s + (4.18 + 0.685i)18-s + (0.531 − 1.17i)22-s + (−4.58 + 2.64i)23-s + (−4.33 − 2.5i)25-s + (4.29 + 4.29i)29-s + (5.42 + 1.60i)32-s + (−3.96 − 4.49i)36-s + (−11.3 − 3.03i)37-s + (−8.64 + 8.64i)43-s + (−1.63 + 0.812i)44-s + ⋯
L(s)  = 1  + (−0.773 − 0.633i)2-s + (0.197 + 0.980i)4-s + (0.467 − 0.883i)8-s + (−0.866 + 0.5i)9-s + (0.0712 + 0.265i)11-s + (−0.921 + 0.387i)16-s + (0.986 + 0.161i)18-s + (0.113 − 0.250i)22-s + (−0.955 + 0.551i)23-s + (−0.866 − 0.5i)25-s + (0.796 + 0.796i)29-s + (0.958 + 0.283i)32-s + (−0.661 − 0.749i)36-s + (−1.86 − 0.498i)37-s + (−1.31 + 1.31i)43-s + (−0.246 + 0.122i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.383 - 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.383 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $-0.383 - 0.923i$
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: \((2,\ 784,\ (\ :1/2),\ -0.383 - 0.923i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.196459 + 0.294475i\)
\(L(\frac12)\) \(\approx\) \(0.196459 + 0.294475i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.09 + 0.895i)T \)
7 \( 1 \)
good3 \( 1 + (2.59 - 1.5i)T^{2} \)
5 \( 1 + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (-0.236 - 0.882i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + 13iT^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (4.58 - 2.64i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.29 - 4.29i)T + 29iT^{2} \)
31 \( 1 + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (11.3 + 3.03i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + (8.64 - 8.64i)T - 43iT^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.106 - 0.398i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (13.5 - 3.63i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 16iT - 71T^{2} \)
73 \( 1 + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.93 - 13.7i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 83iT^{2} \)
89 \( 1 + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 97T^{2} \)
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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

−10.42787075675501217939082838198, −9.860826745495382832498348233204, −8.817732804317628149782645840750, −8.221541197423832872892610960023, −7.38118858658597518198750198740, −6.33728347323035539313183204409, −5.14086067337695856430986730129, −3.90211404591025880992056634188, −2.82420582949131647399168049493, −1.70276952447676520637939884974, 0.21849073398598007998131192210, 1.95897681823047735379681378195, 3.42401684698376559474910332538, 4.87164397695977601278220876401, 5.89640213200444190815423366214, 6.48731980748397413569236223628, 7.55979867946836848841482272732, 8.431482621113110184355854774041, 8.966532014004152427629795195570, 9.965829574376017905636710771352

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