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Properties

Label 2-700-140.59-c1-0-40
Degree $2$
Conductor $700$
Sign $-0.738 + 0.673i$
Analytic cond. $5.58952$
Root an. cond. $2.36421$
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  + (−0.620 − 1.27i)2-s + (0.573 + 0.331i)3-s + (−1.23 + 1.57i)4-s + (0.0650 − 0.934i)6-s + (−2.03 + 1.68i)7-s + (2.76 + 0.585i)8-s + (−1.28 − 2.21i)9-s + (−3.12 − 1.80i)11-s + (−1.22 + 0.496i)12-s + 5.83·13-s + (3.40 + 1.54i)14-s + (−0.971 − 3.88i)16-s + (0.684 − 1.18i)17-s + (−2.02 + 3.00i)18-s + (−2.04 − 3.54i)19-s + ⋯
L(s)  = 1  + (−0.438 − 0.898i)2-s + (0.331 + 0.191i)3-s + (−0.615 + 0.788i)4-s + (0.0265 − 0.381i)6-s + (−0.770 + 0.637i)7-s + (0.978 + 0.207i)8-s + (−0.426 − 0.739i)9-s + (−0.943 − 0.544i)11-s + (−0.354 + 0.143i)12-s + 1.61·13-s + (0.911 + 0.412i)14-s + (−0.242 − 0.970i)16-s + (0.165 − 0.287i)17-s + (−0.477 + 0.707i)18-s + (−0.469 − 0.813i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.738 + 0.673i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.738 + 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.738 + 0.673i$
Analytic conductor: \(5.58952\)
Root analytic conductor: \(2.36421\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{700} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 700,\ (\ :1/2),\ -0.738 + 0.673i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.285729 - 0.737475i\)
\(L(\frac12)\) \(\approx\) \(0.285729 - 0.737475i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.620 + 1.27i)T \)
5 \( 1 \)
7 \( 1 + (2.03 - 1.68i)T \)
good3 \( 1 + (-0.573 - 0.331i)T + (1.5 + 2.59i)T^{2} \)
11 \( 1 + (3.12 + 1.80i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 5.83T + 13T^{2} \)
17 \( 1 + (-0.684 + 1.18i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.04 + 3.54i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.62 + 2.81i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 5.19T + 29T^{2} \)
31 \( 1 + (-4.43 + 7.67i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-9.34 + 5.39i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 0.832iT - 41T^{2} \)
43 \( 1 + 3.10T + 43T^{2} \)
47 \( 1 + (5.97 - 3.44i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.42 + 3.70i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.73 + 6.47i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.28 + 0.742i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.26 + 2.19i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 3.52iT - 71T^{2} \)
73 \( 1 + (2.58 - 4.47i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (9.82 - 5.67i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 6.49iT - 83T^{2} \)
89 \( 1 + (8.13 - 4.69i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 0.343T + 97T^{2} \)
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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

−9.968106652607310298697819688357, −9.344123424437625998849397204821, −8.574888188759009550894524100691, −8.011508001712259674764775494879, −6.49010927264666948406840380177, −5.65013881125378594423071659042, −4.14041216721309526865843081583, −3.22935961226930815207683154368, −2.44178935386521456764638294788, −0.47115492514589251852561660454, 1.55388527622813467851299932056, 3.29165165744332642723050566868, 4.49379686767415125941674438980, 5.68981677399078270175759229384, 6.39712973245534738703483324314, 7.46715655163999318036708260363, 8.097128222365775733343409713842, 8.785832420454092609994871072873, 9.961062148349370251789283395818, 10.42890295161667587150906133402

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