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Properties

Label 2-700-100.23-c1-0-28
Degree $2$
Conductor $700$
Sign $-0.350 - 0.936i$
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  + (−0.0542 + 1.41i)2-s + (0.700 + 1.37i)3-s + (−1.99 − 0.153i)4-s + (0.861 − 2.06i)5-s + (−1.98 + 0.915i)6-s + (0.707 + 0.707i)7-s + (0.324 − 2.80i)8-s + (0.362 − 0.499i)9-s + (2.86 + 1.32i)10-s + (−0.460 − 0.633i)11-s + (−1.18 − 2.85i)12-s + (0.683 + 4.31i)13-s + (−1.03 + 0.960i)14-s + (3.44 − 0.261i)15-s + (3.95 + 0.611i)16-s + (2.27 + 1.15i)17-s + ⋯
L(s)  = 1  + (−0.0383 + 0.999i)2-s + (0.404 + 0.794i)3-s + (−0.997 − 0.0766i)4-s + (0.385 − 0.922i)5-s + (−0.809 + 0.373i)6-s + (0.267 + 0.267i)7-s + (0.114 − 0.993i)8-s + (0.120 − 0.166i)9-s + (0.907 + 0.420i)10-s + (−0.138 − 0.191i)11-s + (−0.342 − 0.822i)12-s + (0.189 + 1.19i)13-s + (−0.277 + 0.256i)14-s + (0.888 − 0.0675i)15-s + (0.988 + 0.152i)16-s + (0.551 + 0.280i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.350 - 0.936i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.350 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.350 - 0.936i$
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: \((2,\ 700,\ (\ :1/2),\ -0.350 - 0.936i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.981045 + 1.41415i\)
\(L(\frac12)\) \(\approx\) \(0.981045 + 1.41415i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.0542 - 1.41i)T \)
5 \( 1 + (-0.861 + 2.06i)T \)
7 \( 1 + (-0.707 - 0.707i)T \)
good3 \( 1 + (-0.700 - 1.37i)T + (-1.76 + 2.42i)T^{2} \)
11 \( 1 + (0.460 + 0.633i)T + (-3.39 + 10.4i)T^{2} \)
13 \( 1 + (-0.683 - 4.31i)T + (-12.3 + 4.01i)T^{2} \)
17 \( 1 + (-2.27 - 1.15i)T + (9.99 + 13.7i)T^{2} \)
19 \( 1 + (-2.34 - 7.21i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (1.03 - 6.54i)T + (-21.8 - 7.10i)T^{2} \)
29 \( 1 + (4.49 + 1.46i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (-4.02 + 1.30i)T + (25.0 - 18.2i)T^{2} \)
37 \( 1 + (-6.40 + 1.01i)T + (35.1 - 11.4i)T^{2} \)
41 \( 1 + (5.06 + 3.67i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + (-3.22 + 3.22i)T - 43iT^{2} \)
47 \( 1 + (-4.56 + 2.32i)T + (27.6 - 38.0i)T^{2} \)
53 \( 1 + (-3.28 + 1.67i)T + (31.1 - 42.8i)T^{2} \)
59 \( 1 + (-1.50 - 1.09i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-1.54 + 1.11i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-1.82 + 3.58i)T + (-39.3 - 54.2i)T^{2} \)
71 \( 1 + (3.85 + 1.25i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (5.29 + 0.838i)T + (69.4 + 22.5i)T^{2} \)
79 \( 1 + (2.14 - 6.59i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (12.3 + 6.29i)T + (48.7 + 67.1i)T^{2} \)
89 \( 1 + (10.0 + 13.8i)T + (-27.5 + 84.6i)T^{2} \)
97 \( 1 + (2.21 + 4.35i)T + (-57.0 + 78.4i)T^{2} \)
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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

−10.12640015304078531315493841460, −9.661521178266962295604708997970, −8.966218173683574735897687911390, −8.264346089281023479172775023924, −7.36389592770271952940138768239, −6.00413738704501609628618481958, −5.44996374964115735358825214317, −4.30460376367594668416868649215, −3.69319530515933181481689121356, −1.49136071983726130194373103681, 1.04666435335587113896013540318, 2.49608311949131386972635356710, 2.98619863294017685251351951248, 4.49478864961232242275022297538, 5.57285758991460460768655595029, 6.88001660241955788958155815064, 7.67756315963007649577709711518, 8.415664639924523303831385351519, 9.563427421418900913651375293801, 10.35618280638795988401836152637

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