Properties

Label 1-35-35.23-r1-0-0
Degree $1$
Conductor $35$
Sign $-0.0333 + 0.999i$
Analytic cond. $3.76127$
Root an. cond. $3.76127$
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  + (−0.866 + 0.5i)2-s + (−0.866 − 0.5i)3-s + (0.5 − 0.866i)4-s + 6-s + i·8-s + (0.5 + 0.866i)9-s + (−0.5 + 0.866i)11-s + (−0.866 + 0.5i)12-s + i·13-s + (−0.5 − 0.866i)16-s + (0.866 + 0.5i)17-s + (−0.866 − 0.5i)18-s + (0.5 + 0.866i)19-s i·22-s + (0.866 − 0.5i)23-s + (0.5 − 0.866i)24-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)2-s + (−0.866 − 0.5i)3-s + (0.5 − 0.866i)4-s + 6-s + i·8-s + (0.5 + 0.866i)9-s + (−0.5 + 0.866i)11-s + (−0.866 + 0.5i)12-s + i·13-s + (−0.5 − 0.866i)16-s + (0.866 + 0.5i)17-s + (−0.866 − 0.5i)18-s + (0.5 + 0.866i)19-s i·22-s + (0.866 − 0.5i)23-s + (0.5 − 0.866i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0333 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0333 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(35\)    =    \(5 \cdot 7\)
Sign: $-0.0333 + 0.999i$
Analytic conductor: \(3.76127\)
Root analytic conductor: \(3.76127\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{35} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 35,\ (1:\ ),\ -0.0333 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3940914981 + 0.4074788136i\)
\(L(\frac12)\) \(\approx\) \(0.3940914981 + 0.4074788136i\)
\(L(1)\) \(\approx\) \(0.5259784365 + 0.1599414237i\)
\(L(1)\) \(\approx\) \(0.5259784365 + 0.1599414237i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 + (-0.866 - 0.5i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + iT \)
17 \( 1 + (0.866 + 0.5i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (0.866 - 0.5i)T \)
29 \( 1 - T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (-0.866 + 0.5i)T \)
41 \( 1 + T \)
43 \( 1 + iT \)
47 \( 1 + (-0.866 + 0.5i)T \)
53 \( 1 + (-0.866 - 0.5i)T \)
59 \( 1 + (0.5 - 0.866i)T \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + (0.866 + 0.5i)T \)
71 \( 1 + T \)
73 \( 1 + (-0.866 - 0.5i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + iT \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 - iT \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−35.11772007639093372883908462483, −34.56691511976696396446704353912, −33.26638670852837120971248172461, −31.77444632416938725784851753182, −29.99110452644838179159717983775, −29.19033676384501073990784057037, −27.98034907519104417487522402386, −27.15748830477500801629446399680, −25.99300691800533879507516452989, −24.3786396986232881309157891134, −22.732413352769022420498266007834, −21.52356671931436109022274071222, −20.48659491098409147068331501989, −18.83589264824499655215871770103, −17.71511232749917223259956547541, −16.56947069915554402805379144156, −15.47776606501942698877079110925, −12.995570693817099458925329474223, −11.53325282543335546082723962884, −10.56092859611644771217775593819, −9.23858059425651168162544138315, −7.45115921623780644906790709824, −5.51761052610238754405018686488, −3.300405893387460143152213144727, −0.58660318278782783810342301815, 1.614142252577517778369903397108, 5.17305827598066184261186499461, 6.651427155432718658422555601447, 7.836761687865052724542143333437, 9.74251264738153161888673278144, 11.06255783167064549152496437350, 12.5061863171024189798533139335, 14.46249130538229529448216438267, 16.10979359357073578779150555585, 17.08237789926625500570762941508, 18.28167289380249991367582095931, 19.189740522871831255464499885724, 20.91736057201061889718337743494, 22.84792428868406748886644247061, 23.79069625323264199036525808920, 24.95152094846234389502100099983, 26.20805278624132193919093484434, 27.636606319691167828325626589449, 28.58087648188883751316775995386, 29.48835474049905042769289033941, 31.03724032898790829925069450985, 32.9734071491104142330023578203, 33.84810407164582833472657257167, 34.81779179651389218949803737421, 35.928886150869338157071003557969

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