Properties

Label 2-700-100.23-c1-0-27
Degree $2$
Conductor $700$
Sign $-0.428 - 0.903i$
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  + (0.931 + 1.06i)2-s + (−0.268 − 0.527i)3-s + (−0.265 + 1.98i)4-s + (0.994 + 2.00i)5-s + (0.311 − 0.777i)6-s + (−0.707 − 0.707i)7-s + (−2.35 + 1.56i)8-s + (1.55 − 2.14i)9-s + (−1.20 + 2.92i)10-s + (1.31 + 1.81i)11-s + (1.11 − 0.392i)12-s + (0.761 + 4.80i)13-s + (0.0942 − 1.41i)14-s + (0.789 − 1.06i)15-s + (−3.85 − 1.05i)16-s + (3.70 + 1.88i)17-s + ⋯
L(s)  = 1  + (0.658 + 0.752i)2-s + (−0.155 − 0.304i)3-s + (−0.132 + 0.991i)4-s + (0.444 + 0.895i)5-s + (0.127 − 0.317i)6-s + (−0.267 − 0.267i)7-s + (−0.833 + 0.552i)8-s + (0.519 − 0.714i)9-s + (−0.381 + 0.924i)10-s + (0.397 + 0.546i)11-s + (0.322 − 0.113i)12-s + (0.211 + 1.33i)13-s + (0.0251 − 0.377i)14-s + (0.203 − 0.274i)15-s + (−0.964 − 0.263i)16-s + (0.897 + 0.457i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.428 - 0.903i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.428 - 0.903i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.428 - 0.903i$
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: \((2,\ 700,\ (\ :1/2),\ -0.428 - 0.903i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.10235 + 1.74217i\)
\(L(\frac12)\) \(\approx\) \(1.10235 + 1.74217i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.931 - 1.06i)T \)
5 \( 1 + (-0.994 - 2.00i)T \)
7 \( 1 + (0.707 + 0.707i)T \)
good3 \( 1 + (0.268 + 0.527i)T + (-1.76 + 2.42i)T^{2} \)
11 \( 1 + (-1.31 - 1.81i)T + (-3.39 + 10.4i)T^{2} \)
13 \( 1 + (-0.761 - 4.80i)T + (-12.3 + 4.01i)T^{2} \)
17 \( 1 + (-3.70 - 1.88i)T + (9.99 + 13.7i)T^{2} \)
19 \( 1 + (1.35 + 4.16i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (1.13 - 7.19i)T + (-21.8 - 7.10i)T^{2} \)
29 \( 1 + (4.52 + 1.46i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (3.25 - 1.05i)T + (25.0 - 18.2i)T^{2} \)
37 \( 1 + (-0.224 + 0.0355i)T + (35.1 - 11.4i)T^{2} \)
41 \( 1 + (-5.11 - 3.71i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + (-7.56 + 7.56i)T - 43iT^{2} \)
47 \( 1 + (0.175 - 0.0892i)T + (27.6 - 38.0i)T^{2} \)
53 \( 1 + (-11.1 + 5.68i)T + (31.1 - 42.8i)T^{2} \)
59 \( 1 + (2.11 + 1.53i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (9.73 - 7.07i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-6.74 + 13.2i)T + (-39.3 - 54.2i)T^{2} \)
71 \( 1 + (2.08 + 0.678i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (0.792 + 0.125i)T + (69.4 + 22.5i)T^{2} \)
79 \( 1 + (-1.50 + 4.61i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-14.2 - 7.25i)T + (48.7 + 67.1i)T^{2} \)
89 \( 1 + (-1.36 - 1.87i)T + (-27.5 + 84.6i)T^{2} \)
97 \( 1 + (-4.59 - 9.00i)T + (-57.0 + 78.4i)T^{2} \)
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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

−10.88400331554254355473617161226, −9.552580944590006085650655841084, −9.200044211317184526349811820772, −7.58017658230133434253229910623, −7.07033191579626209083430873990, −6.40362271721242299730846708032, −5.61207106627994385523106262962, −4.16234485940324520811805905407, −3.49594170221833656300214037133, −1.94284240057086774033346757006, 0.948899524964243198415916499924, 2.34625718760336043662065863091, 3.64959348976049959243823119399, 4.62243847098948275976439212964, 5.61477181971826436642925229312, 5.99207514812747363156745100708, 7.66053577502652378073720184584, 8.698653055477938274962700150486, 9.570884891500499565496175131374, 10.32721611006028649608433747183

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