Properties

Label 2-700-100.23-c1-0-1
Degree $2$
Conductor $700$
Sign $-0.493 - 0.869i$
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  + (0.527 + 1.31i)2-s + (−1.24 − 2.44i)3-s + (−1.44 + 1.38i)4-s + (−1.65 − 1.50i)5-s + (2.55 − 2.92i)6-s + (−0.707 − 0.707i)7-s + (−2.57 − 1.16i)8-s + (−2.66 + 3.66i)9-s + (1.09 − 2.96i)10-s + (−2.05 − 2.83i)11-s + (5.18 + 1.80i)12-s + (0.978 + 6.18i)13-s + (0.554 − 1.30i)14-s + (−1.61 + 5.92i)15-s + (0.162 − 3.99i)16-s + (2.62 + 1.33i)17-s + ⋯
L(s)  = 1  + (0.373 + 0.927i)2-s + (−0.719 − 1.41i)3-s + (−0.721 + 0.692i)4-s + (−0.740 − 0.671i)5-s + (1.04 − 1.19i)6-s + (−0.267 − 0.267i)7-s + (−0.911 − 0.410i)8-s + (−0.887 + 1.22i)9-s + (0.346 − 0.937i)10-s + (−0.620 − 0.854i)11-s + (1.49 + 0.520i)12-s + (0.271 + 1.71i)13-s + (0.148 − 0.347i)14-s + (−0.415 + 1.52i)15-s + (0.0405 − 0.999i)16-s + (0.636 + 0.324i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.493 - 0.869i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.493 - 0.869i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.493 - 0.869i$
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: \((2,\ 700,\ (\ :1/2),\ -0.493 - 0.869i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.210886 + 0.362366i\)
\(L(\frac12)\) \(\approx\) \(0.210886 + 0.362366i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.527 - 1.31i)T \)
5 \( 1 + (1.65 + 1.50i)T \)
7 \( 1 + (0.707 + 0.707i)T \)
good3 \( 1 + (1.24 + 2.44i)T + (-1.76 + 2.42i)T^{2} \)
11 \( 1 + (2.05 + 2.83i)T + (-3.39 + 10.4i)T^{2} \)
13 \( 1 + (-0.978 - 6.18i)T + (-12.3 + 4.01i)T^{2} \)
17 \( 1 + (-2.62 - 1.33i)T + (9.99 + 13.7i)T^{2} \)
19 \( 1 + (-1.98 - 6.10i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (0.192 - 1.21i)T + (-21.8 - 7.10i)T^{2} \)
29 \( 1 + (2.17 + 0.707i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (-8.96 + 2.91i)T + (25.0 - 18.2i)T^{2} \)
37 \( 1 + (9.22 - 1.46i)T + (35.1 - 11.4i)T^{2} \)
41 \( 1 + (-2.24 - 1.63i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + (-1.19 + 1.19i)T - 43iT^{2} \)
47 \( 1 + (6.22 - 3.17i)T + (27.6 - 38.0i)T^{2} \)
53 \( 1 + (8.46 - 4.31i)T + (31.1 - 42.8i)T^{2} \)
59 \( 1 + (5.93 + 4.31i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (3.53 - 2.56i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (4.51 - 8.86i)T + (-39.3 - 54.2i)T^{2} \)
71 \( 1 + (5.52 + 1.79i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (1.32 + 0.209i)T + (69.4 + 22.5i)T^{2} \)
79 \( 1 + (2.06 - 6.36i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (1.66 + 0.849i)T + (48.7 + 67.1i)T^{2} \)
89 \( 1 + (-5.37 - 7.40i)T + (-27.5 + 84.6i)T^{2} \)
97 \( 1 + (-4.85 - 9.52i)T + (-57.0 + 78.4i)T^{2} \)
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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

−11.13286061104059638775253264141, −9.647819195699155200241291005954, −8.484093547268973432158173162555, −7.910029729014786585024639528134, −7.21692828155965890596055617256, −6.27664558423476706705408772689, −5.70073330573981247467251526386, −4.53953671000452576113849848511, −3.42929357486290451111544238851, −1.32483695799204146737158194801, 0.23335558017784438930991479570, 2.89163299719238835092846182432, 3.41434382351635171652999518060, 4.71459991830837501563921516581, 5.14216141697326581893056877062, 6.22451530038851024428964697407, 7.61428967117920650810866725764, 8.799580557152344576607960826132, 9.847633598608872262847947579740, 10.35034188350193494413934782283

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