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Properties

Label 2-700-140.27-c0-0-4
Degree $2$
Conductor $700$
Sign $0.755 - 0.655i$
Analytic cond. $0.349345$
Root an. cond. $0.591054$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.965 + 0.258i)2-s + (0.866 + 0.499i)4-s + (−0.707 + 0.707i)7-s + (0.707 + 0.707i)8-s + i·9-s − 1.73i·11-s + (−0.866 + 0.500i)14-s + (0.500 + 0.866i)16-s + (−0.258 + 0.965i)18-s + (0.448 − 1.67i)22-s + (−0.707 − 0.707i)23-s + (−0.965 + 0.258i)28-s − i·29-s + (0.258 + 0.965i)32-s + (−0.499 + 0.866i)36-s + (−1.22 − 1.22i)37-s + ⋯
L(s)  = 1  + (0.965 + 0.258i)2-s + (0.866 + 0.499i)4-s + (−0.707 + 0.707i)7-s + (0.707 + 0.707i)8-s + i·9-s − 1.73i·11-s + (−0.866 + 0.500i)14-s + (0.500 + 0.866i)16-s + (−0.258 + 0.965i)18-s + (0.448 − 1.67i)22-s + (−0.707 − 0.707i)23-s + (−0.965 + 0.258i)28-s − i·29-s + (0.258 + 0.965i)32-s + (−0.499 + 0.866i)36-s + (−1.22 − 1.22i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 - 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 - 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.755 - 0.655i$
Analytic conductor: \(0.349345\)
Root analytic conductor: \(0.591054\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{700} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 700,\ (\ :0),\ 0.755 - 0.655i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.560325594\)
\(L(\frac12)\) \(\approx\) \(1.560325594\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 - 0.258i)T \)
5 \( 1 \)
7 \( 1 + (0.707 - 0.707i)T \)
good3 \( 1 - iT^{2} \)
11 \( 1 + 1.73iT - T^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
29 \( 1 + iT - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + (1.22 + 1.22i)T + iT^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
71 \( 1 - 1.73iT - T^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 + 1.73T + T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 - iT^{2} \)
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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

−10.97968503070652754690995185834, −10.11417448359043539827407574309, −8.739381541905003617148635487272, −8.162662026463814848009533516129, −7.07761368224142853960767539689, −5.92385767924668253678910722828, −5.65221649879381497239269508450, −4.32642430177412851528449502369, −3.21329761982355484730702655539, −2.30178237339360672553445943108, 1.66943390920565108435657919519, 3.17732591937864372948970145288, 4.00914791672696202562444599787, 4.91991764935703888691052184219, 6.13927779952595730749466896624, 6.93176737485648561182155614529, 7.49106585464928939504090449368, 9.147069223303377598388159742820, 9.974970296394776386547990565951, 10.44437780102103021253732854981

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