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Properties

Label 2-700-100.23-c1-0-44
Degree $2$
Conductor $700$
Sign $0.982 + 0.185i$
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  + (0.0492 + 1.41i)2-s + (0.00732 + 0.0143i)3-s + (−1.99 + 0.139i)4-s + (−2.23 + 0.100i)5-s + (−0.0199 + 0.0110i)6-s + (0.707 + 0.707i)7-s + (−0.295 − 2.81i)8-s + (1.76 − 2.42i)9-s + (−0.251 − 3.15i)10-s + (−1.01 − 1.39i)11-s + (−0.0166 − 0.0276i)12-s + (0.349 + 2.20i)13-s + (−0.964 + 1.03i)14-s + (−0.0178 − 0.0313i)15-s + (3.96 − 0.556i)16-s + (−2.42 − 1.23i)17-s + ⋯
L(s)  = 1  + (0.0348 + 0.999i)2-s + (0.00423 + 0.00830i)3-s + (−0.997 + 0.0696i)4-s + (−0.998 + 0.0448i)5-s + (−0.00815 + 0.00451i)6-s + (0.267 + 0.267i)7-s + (−0.104 − 0.994i)8-s + (0.587 − 0.808i)9-s + (−0.0796 − 0.996i)10-s + (−0.306 − 0.421i)11-s + (−0.00479 − 0.00798i)12-s + (0.0969 + 0.612i)13-s + (−0.257 + 0.276i)14-s + (−0.00459 − 0.00810i)15-s + (0.990 − 0.139i)16-s + (−0.588 − 0.299i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 + 0.185i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.982 + 0.185i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.982 + 0.185i$
Analytic conductor: \(5.58952\)
Root analytic conductor: \(2.36421\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{700} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 700,\ (\ :1/2),\ 0.982 + 0.185i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.941560 - 0.0878614i\)
\(L(\frac12)\) \(\approx\) \(0.941560 - 0.0878614i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0492 - 1.41i)T \)
5 \( 1 + (2.23 - 0.100i)T \)
7 \( 1 + (-0.707 - 0.707i)T \)
good3 \( 1 + (-0.00732 - 0.0143i)T + (-1.76 + 2.42i)T^{2} \)
11 \( 1 + (1.01 + 1.39i)T + (-3.39 + 10.4i)T^{2} \)
13 \( 1 + (-0.349 - 2.20i)T + (-12.3 + 4.01i)T^{2} \)
17 \( 1 + (2.42 + 1.23i)T + (9.99 + 13.7i)T^{2} \)
19 \( 1 + (2.57 + 7.91i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-0.352 + 2.22i)T + (-21.8 - 7.10i)T^{2} \)
29 \( 1 + (-3.47 - 1.12i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (-9.19 + 2.98i)T + (25.0 - 18.2i)T^{2} \)
37 \( 1 + (3.89 - 0.617i)T + (35.1 - 11.4i)T^{2} \)
41 \( 1 + (-6.44 - 4.68i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + (-3.18 + 3.18i)T - 43iT^{2} \)
47 \( 1 + (0.782 - 0.398i)T + (27.6 - 38.0i)T^{2} \)
53 \( 1 + (-7.79 + 3.97i)T + (31.1 - 42.8i)T^{2} \)
59 \( 1 + (10.9 + 7.93i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (1.02 - 0.745i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-2.89 + 5.67i)T + (-39.3 - 54.2i)T^{2} \)
71 \( 1 + (12.6 + 4.12i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (10.0 + 1.58i)T + (69.4 + 22.5i)T^{2} \)
79 \( 1 + (-3.89 + 11.9i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (10.6 + 5.44i)T + (48.7 + 67.1i)T^{2} \)
89 \( 1 + (1.96 + 2.70i)T + (-27.5 + 84.6i)T^{2} \)
97 \( 1 + (-5.73 - 11.2i)T + (-57.0 + 78.4i)T^{2} \)
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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

−10.34258347593559920580665523576, −9.091554881780926969570323739449, −8.729585859468897621672734124981, −7.72456537608272176972363393426, −6.84653558479755826124520766884, −6.29074833450751879957361411998, −4.72199508105718819508535898809, −4.35021354650832095446071190376, −2.98351926140349421352737442803, −0.55698008875366819172543589124, 1.36727576468456590146785686975, 2.74202297783108532746515873810, 4.06130302865934691032913608488, 4.53528535906098325767400158706, 5.72894243775005172115242593057, 7.29901250823739707473329667776, 8.069600605538014830989731592389, 8.627136619387719592142116350018, 10.09487980616356679309232064462, 10.44155233729755247710887371229

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