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Properties

Label 2-700-35.19-c4-0-44
Degree $2$
Conductor $700$
Sign $-0.999 - 0.0166i$
Analytic cond. $72.3589$
Root an. cond. $8.50640$
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  + (0.252 + 0.437i)3-s + (48.2 − 8.29i)7-s + (40.3 − 69.9i)9-s + (−106. − 183. i)11-s − 45.2·13-s + (125. + 217. i)17-s + (−405. − 234. i)19-s + (15.8 + 19.0i)21-s + (−809. − 467. i)23-s + 81.7·27-s + 816.·29-s + (−853. + 492. i)31-s + (53.6 − 93.0i)33-s + (−474. − 273. i)37-s + (−11.4 − 19.8i)39-s + ⋯
L(s)  = 1  + (0.0280 + 0.0486i)3-s + (0.985 − 0.169i)7-s + (0.498 − 0.863i)9-s + (−0.877 − 1.51i)11-s − 0.267·13-s + (0.435 + 0.753i)17-s + (−1.12 − 0.649i)19-s + (0.0359 + 0.0432i)21-s + (−1.52 − 0.883i)23-s + 0.112·27-s + 0.970·29-s + (−0.888 + 0.512i)31-s + (0.0493 − 0.0854i)33-s + (−0.346 − 0.200i)37-s + (−0.00751 − 0.0130i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0166i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0166i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.999 - 0.0166i$
Analytic conductor: \(72.3589\)
Root analytic conductor: \(8.50640\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{700} (649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 700,\ (\ :2),\ -0.999 - 0.0166i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.6539749042\)
\(L(\frac12)\) \(\approx\) \(0.6539749042\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 + (-48.2 + 8.29i)T \)
good3 \( 1 + (-0.252 - 0.437i)T + (-40.5 + 70.1i)T^{2} \)
11 \( 1 + (106. + 183. i)T + (-7.32e3 + 1.26e4i)T^{2} \)
13 \( 1 + 45.2T + 2.85e4T^{2} \)
17 \( 1 + (-125. - 217. i)T + (-4.17e4 + 7.23e4i)T^{2} \)
19 \( 1 + (405. + 234. i)T + (6.51e4 + 1.12e5i)T^{2} \)
23 \( 1 + (809. + 467. i)T + (1.39e5 + 2.42e5i)T^{2} \)
29 \( 1 - 816.T + 7.07e5T^{2} \)
31 \( 1 + (853. - 492. i)T + (4.61e5 - 7.99e5i)T^{2} \)
37 \( 1 + (474. + 273. i)T + (9.37e5 + 1.62e6i)T^{2} \)
41 \( 1 - 2.48e3iT - 2.82e6T^{2} \)
43 \( 1 - 1.88e3iT - 3.41e6T^{2} \)
47 \( 1 + (289. - 502. i)T + (-2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 + (-312. + 180. i)T + (3.94e6 - 6.83e6i)T^{2} \)
59 \( 1 + (5.28e3 - 3.04e3i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (1.53e3 + 884. i)T + (6.92e6 + 1.19e7i)T^{2} \)
67 \( 1 + (-5.17e3 + 2.98e3i)T + (1.00e7 - 1.74e7i)T^{2} \)
71 \( 1 - 1.01e3T + 2.54e7T^{2} \)
73 \( 1 + (-1.54e3 - 2.67e3i)T + (-1.41e7 + 2.45e7i)T^{2} \)
79 \( 1 + (361. - 626. i)T + (-1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 - 8.10e3T + 4.74e7T^{2} \)
89 \( 1 + (2.39e3 + 1.37e3i)T + (3.13e7 + 5.43e7i)T^{2} \)
97 \( 1 + 1.66e4T + 8.85e7T^{2} \)
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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

−9.404793677081057377230471096318, −8.269941805822597456104022390185, −8.073338295248116279648256774707, −6.65381132398494750412805973188, −5.91168364567639511049584130551, −4.78078565098738053658186782146, −3.87762743283480979419319982791, −2.68922522822876145095283251505, −1.33623318424690748230803426882, −0.14517623425416147251304888355, 1.79316072047752613109972565164, 2.25942138427974275606151525352, 4.05460027612460233674284742648, 4.88570367478690676024550198386, 5.58524394495989462102404095720, 7.08950698458636393712207650342, 7.70329111394387442768828494854, 8.324003014229644731283010638073, 9.636504795890850777885651414445, 10.28561541615328140769309047241

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